RE : bilax monoidal functors

RE : bilax monoidal functors

[John Baez]

André Joyal wrote:

I am using the following terminology for
> higher braided monoidal (higher) categories:
>
> Monoidal< braided < 2-braided <.......<symmetric
>
> A (n+1)-braided n-category is symmetric
> according to your stabilisation hypothesis.
>
> Is this a good terminology?
>

I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided".  This
seems preferable to me, not because it sounds nicer - it doesn't - but
because it starts counting at a somewhat more natural place.  I believe that
counting monoidal structures is more natural than counting braidings.

For example, a doubly monoidal n-category, one with two compatible monoidal
structures, is a braided monoidal n-category.    I believe this is a theorem
proved by you and Ross when n = 1.  This way of thinking clarifies the
relation between braided monoidal categories and double loop spaces.

Various numbers become more complicated when one counts braidings rather
than monoidal structures:

An n-tuply monoidal k-category is (conjecturally) a special sort of
(n+k)-category... while an n-braided category is a special sort of
(n+k+1)-category.

Similarly: n-dimensional surfaces in (n+k)-dimensional space are n-morphisms
in a k-tuply monoidal n-category... but they are n-morphisms in an
(k-1)-braided n-category.

And so on.

On the other hand, if it's braidings that you really want to count, rather
than monoidal structures, your terminology is perfect.

By the way: I don't remember anyone on this mailing list ever asking if
their own terminology is good.  I only remember them complaining about other
people's terminology.  I applaud your departure from this unpleasant
tradition!

Best,
jb


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autonomous terminology: WAS: bilax monoidal functors

[Dusko Pavlovic]

> By the way: I don't remember anyone on this mailing list ever asking if
> their own terminology is good.  I only remember them complaining about other
> people's terminology.  I applaud your departure from this unpleasant
> tradition!

to support this departure, i have a terminology question.

last couple of years *dagger monoidal* and *dagger compact* categories
came to be popular. in a recent paper i encountered lots of star
autonomous categories with an additional dagger structure.

i am reluctant call them dagger star autonomous categories, because it is
a mouthful. moreover it seems that listing the operations of a signature
in its name is a bad naming strategy. trying to maintain descriptive names
is a lost cause. linguists have known that languages are not descriptive
since XIX century. mathematicians since much earlier, even since they
started calling everything x and y. we never try to give cars or people
descriptive names, only mathematical structures. a new chemical element is
given an ugly descriptive name only until a simpler one is agreed upon.

i was going to call them *dagger autonomous* but peter selinger pointed
out that this is confusing. indeed, the term *autonomous* has established
a confusing tradition all on its own:

* i believe that fred linton introduced it in the 60s for what would now
probably be called *closed* structure

* barr followed linton's usage with his star autonomous categories. there
are 10s of 1000s of papers using this terminology (eg from the linear
logic times).

* on the other hand, joyal and street called autonomous those categories
where every object has a monoidal dual. that terminology also caught on.

so now, what should we call those "dagger star autonomous categories" if
we don't want to type 30 characters each time we mention them?

peter suggests DSA-categories. (maybe someone will abbreviate them to
D-categories...)

help appreciated.

-- dusko

PS maybe we should rename dagger monoidal to pink monoidal, and star
autonomous to floyd, so dagger star autonomous categories would be pink
floyd categories.

is there any reason why words should be taken seriously?


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Re: bilax_monoidal_functors?=

[Andre Joyal]

Dear John and Michael,

It all depends on where you start counting.
For americans, the first floor of a buiding is the ground floor
but for most europeans, it is the floor right above: 

http://en.wikipedia.org/wiki/Storey#Numbering

We sometime need to recall in which part of the world we are 
when we take an elevator!
But a ten stories building is the same for everyone.  

More seriously, John wrote:

>I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided".  This
>seems preferable to me, not because it sounds nicer - it doesn't - but
>because it starts counting at a somewhat more natural place.  I believe that
>counting monoidal structures is more natural than counting braidings.

Michael wrote:

>I am using a mixture of your terminologies:
>  monoidal = 1-braided
>  braided = 2-braided
>  sylleptic = 3-braided

I understand your ideas both. Along the same line we could also use:

E1-category = Monoidal  
E2-category = Braided monoidal 
E3-category = .....
.....

John wrote:

>By the way: I don't remember anyone on this mailing list ever asking if
>their own terminology is good.  I only remember them complaining about other
>people's terminology.  I applaud your departure from this unpleasant
>tradition!

My goal is to have a public discussion on terminology.
It can be very difficult to agree upon because
adopting one is like commiting to a rule of law,
to a moral code, possibly to a social code.
There is an emotional and social aspect to this commitment.
There is also a psychological aspect because a terminology
looks natural if you use it long enough (it is a matter of a few days).
I hope that a public discussion can help peoples 
choosing their terminology.

I do think that my terminology for higher braided
monoidal categories is quite good.
Let me say a few things in its defense:

First, it extends naturally a terminology which is used 
by the mathematical community since many decades.
Only a specialist can truly appreciate E(k)-categories or 
k-tuply monoidal categories. Second, a braiding is a commutation 
structure. To call a monoidal category 1-braided is kind of 
confusing because there is no commutation structure 
on a general monoidal category. A monoidal category is 0-braided. 
Third, a n-braided (topological or simplicial) group is exactly what 
you need to describe the homotopy type of an n-connected space (n\geq 1). 


I wonder who introduced the notion of E(n)-space and
the terminology?


Best regards, 
André



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Re: autonomous terminology: WAS: bilax monoidal functors

[Colin McLarty]

2010/5/9 Dusko Pavlovic <Dusko.Pavlovic@comlab.ox.ac.uk>:

Asks

> is there any reason why words should be taken seriously?

That just depends on whether or not you want to be understood by
people who do not already know everything you are going to say.

best, Colin


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Re: bilax monoidal functors

[Urs Schreiber]

> An n-tuply monoidal k-category is (conjecturally) a special sort of
> (n+k)-category

By the way, some progress on this is available from John Francis' and
Jacob Lurie's discussion of k-tuply monoidal (n,1)-categories as those
equipped with a little k-cubes action.

In particular there is a proof of the stabilization hypothesis for
(n,1)-categories this way, and an analog of the May recognition
theorem for parameterized oo-groupoids, i.e. (oo,1)-sheaves.

Some of this is summarized with pointers to references here:

  http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category

Best,
Urs


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Re: autonomous terminology: WAS: bilax monoidal functors

[posina]

> is there any reason why words should be taken seriously?

I'd take words seriously for the simple reason that they are an expression
of concepts with which we reason. I hope that this line of questioning is
not indicative of the future behind us: treating the notion of GRAMMAR
lightly (as in replacing grammar with look-up tables), which is a sign of  a
failure to distinguish between the concepts of PARTICULAR and GENERAL
(contexuality does not rationalize confusing GENERAL with PARTICULARS). The
distinction between GENERAL and PARTICULAR is an inheritance that I am most
proud of and thankful for.

Thank you,
posina


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Re: bilax_monoidal_functors?=

[Eduardo J. Dubuc]

Andre points out:

"To call a monoidal category 1-braided is kind of
confusing because there is no commutation structure
on a general monoidal category. A monoidal category is 0-braided."

Being an outsider, with no previous neither usage or opinion on this 
terminology beyond just monoidal and/or tensor category, this seems to 
me definitive, and more than enough to settle the question.

e.d.






Andre Joyal wrote:
> Dear John and Michael,
> 
> It all depends on where you start counting.
> For americans, the first floor of a buiding is the ground floor
> but for most europeans, it is the floor right above: 
> 
> http://en.wikipedia.org/wiki/Storey#Numbering
> 
> We sometime need to recall in which part of the world we are 
> when we take an elevator!
> But a ten stories building is the same for everyone.  
> 
> More seriously, John wrote:
> 
>> I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided".  This
>> seems preferable to me, not because it sounds nicer - it doesn't - but
>> because it starts counting at a somewhat more natural place.  I believe that
>> counting monoidal structures is more natural than counting braidings.
> 
> Michael wrote:
> 
>> I am using a mixture of your terminologies:
>>  monoidal = 1-braided
>>  braided = 2-braided
>>  sylleptic = 3-braided
> 
> I understand your ideas both. Along the same line we could also use:
> 
> E1-category = Monoidal  
> E2-category = Braided monoidal 
> E3-category = .....
> .....
> 
> John wrote:
> 
>> By the way: I don't remember anyone on this mailing list ever asking if
>> their own terminology is good.  I only remember them complaining about  other
>> people's terminology.  I applaud your departure from this unpleasant
>> tradition!
> 
> My goal is to have a public discussion on terminology.
> It can be very difficult to agree upon because
> adopting one is like commiting to a rule of law,
> to a moral code, possibly to a social code.
> There is an emotional and social aspect to this commitment.
> There is also a psychological aspect because a terminology
> looks natural if you use it long enough (it is a matter of a few days).
> I hope that a public discussion can help peoples 
> choosing their terminology.
> 
> I do think that my terminology for higher braided
> monoidal categories is quite good.
> Let me say a few things in its defense:
> 
> First, it extends naturally a terminology which is used 
> by the mathematical community since many decades.
> Only a specialist can truly appreciate E(k)-categories or 
> k-tuply monoidal categories. Second, a braiding is a commutation 
> structure. To call a monoidal category 1-braided is kind of 
> confusing because there is no commutation structure 
> on a general monoidal category. A monoidal category is 0-braided. 
> Third, a n-braided (topological or simplicial) group is exactly what 
> you need to describe the homotopy type of an n-connected space (n\geq 1). 
> 
> 
> I wonder who introduced the notion of E(n)-space and
> the terminology?
> 
> 
> Best regards, 
> André
> 
> 
> 
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Re: autonomous terminology: WAS: bilax monoidal functors

[Jeff Egger]

Hi Dusko,

> i am reluctant call them dagger star autonomous categories,
> because it is a mouthful.

Perhaps it's a symptom of growing up in a country where
"Kangiqsualujjuaq" is considered a perfectly acceptable
name for a village, but I don't think that "dagger star-
autonomous" is a mouthful.  It's only one syllable longer
than "sesquipedalian", and one less than "linearly
distributive", neither of which I would hesitate to use
in day-to-day conversation, should the occasion arise.

It even scans nicely.

Moreover, it communicates something (at least to me); for
better or worse, both "dagger" and "star-autonomous" are
both established terms, and I can see how they might be
combined.  Agglutination, though often mocked, is often
effective.

> so now, what should we call those "dagger star autonomous
> categories" if
> we don't want to type 30 characters each time we mention
> them?

One of the many curious features of the English language is
that adjectives are never inflected; assuming you use TeX,
why not take advantage of this fact in your source code?
\def\dsa{dagger star-autonomous}

> peter suggests DSA-categories.

If you're publishing in a print journal, or a conference
proceedings with a hard page-limit, then that seems sensible
(though I'd drop the hyphen).  Otherwise, do us all a favour
and stick to the long form: pixels are cheap, as editors of
TAC are wont to say.

> (maybe someone will abbreviate them to D-categories...)

What's the point of that?  D-category could stand for (just
plain old) dagger category, or differential category, or any
number of other things.  But maybe someone some day will
\def\dsa{Pavlovic}.

Cheers,
Jeff.





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Re: bilax_monoidal_functors?=

[John Baez]

Eduardo wrote:


> Andre points out:
>
> "To call a monoidal category 1-braided is kind of confusing because there
> is no commutation structure on a general monoidal category. A monoidal
> category is 0-braided."
>
> Being an outsider, with no previous neither usage or opinion on this
> terminology beyond just monoidal and/or tensor category, this seems to me
> definitive, and more than enough to settle the question.


I'm glad that's enough to convince you that Michael Batanin's terminology
"monoidal = 1-braided" is inferior to Andre's "monoidal = 0-braided".

But I think "braided = doubly monoidal" is even better.  After all, a
monoidal category has one tensor product; a braided monoidal category has
two compatible tensor products, and a symmetric monoidal category has three.


But I will not lose sleep if Andre uses "k-braided" as a synonym for
"(k+1)-tuply monoidal".  I don't see it causing any confusion. I just think
it will create more +1's in various formulas.  E.g.: the classifying space
of a k-braided n-category is a (k+1)-fold loop space.

Best,
jb


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Re: bilax_monoidal_functors

[Jeff Egger]

Dear Andre,

> My goal is to have a public discussion on terminology.

It is good that you provoke us into having such discussions!

> It can be very difficult to agree upon because
> adopting one is like commiting to a rule of law,
> to a moral code, possibly to a social code.
> There is an emotional and social aspect to this
> commitment.

I don't understand this at all.  A co-author of mine
recently commented (complained?) that I seem to "change
[my] notation as often as [my] underwear"; and I am not
that much better with terminology.  Indeed, I am overtly
anarchist in this respect, and instinctively resist all
attempts at codifying language.  Most people would agree
that the most important concepts deserve the shortest
names; but people frequently (honestly) disagree over
which concept is the most important.  More significantly,
attitudes often change with time!  It is frustrating,
then, that people will cling to archaic terminology for
the sake of an emotional and social commitment.

[A wonderful counter-example to this phenomenon is when
Mike Barr gave his opinion that the meaning of star-
autonomous category, which initially included symmetry,
should not do so.  I should also say that I think young
mathematicians are generally worse at this than older
ones.  Indeed, the most extreme version of (what I
perceive to be) the same phenomenon is that of the
undergrad who cannot differentiate z=t^2 "because there
is no x".]

My objection to the phrase "autonomous category" (which
Dusko brought up) has less to do with defending Fred
Linton's original usage of that phrase than the fact
that "autonomous category" is a special case (and, from
one point of view, a rather uninteresting special case)
of "star-autonomous category", whereas it sounds like
"star-autonomous category" should mean an "autonomous
category" with some extra structure.  (And, of course,
this once was the case, w.r.t. the older terminology.)
This is confusing; hence one term or the other should
be changed.  I am, in fact, open to all suggestions,
though I cannot help but prefer that "star-autonomous"
be kept and "autonomous" changed.

Cheers,
Jeff.

P.S. A propos of your first email in this thread, why
bother with all those "lax"s?  If you used

> 1) strong monoidal
> 2) monoidal
> 3) comonoidal
> 4) bimonoidal

instead, then you would have

> A monoid is a monoidal functor 1-->C,
> a comonoid is a comonoidal functor 1-->C
> and a bimonoid is a bimonoidal functor 1-->C.

and you could even substitute
> 5) ambimonoidal
for "Frobenius", since "ambialgebra" has been used for
"Frobenius algebra".

Dare I point out that a strong monoidal functor 1-->C
is a _trivial_ monoid?  ;)





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Re: bilax_monoidal_functors?=

[Michael Shulman]

I think it is the least confusing for everyone if when "foo"s start
being decorated with numbers, a "1-foo" is the same thing as what an
unadorned "foo" used to be.  So I definitely have to agree that an
ordinary braided monoidal category should be called "1-braided" if the
naming scheme is going to go by decorating "braided" with numbers.

On the other hand, occasionally it seems to happen that after "foo"s
have been studied for a while, someone introduces a categorified "foo"
and calls it a "bar," and then later someone else comes along and
categorifies again but now starts introducing numbers with "2-bar,"
"3-bar," and so on.  So what really should have been called a "2-foo"
is called a "bar," what really should have been called a "3-foo" is
called a "2-bar," and so on with the numbers all off by one.  As John
points out, the use of "braided = 1-braided" and then "2-braided,"
etc. could be viewed this way, with "monoidal" as the basic "foo" that
we should have started numbering at.

(One other example of this that comes to mind is the original use of
"stack" to mean essentially "2-sheaf," leading to "2-stack" for
something that is really a 3-categorical object, and so on.
Fortunately this particular trend seems to be reversing somewhat.)

However, in the case at hand, it seems to me that there is also an
advantage to the term "braided" over "doubly monoidal."  To give a
category a braided monoidal structure may be *equivalent* to giving it
two interchanging monoidal structures, but that's only true because in
the latter case, the interchange law forces the two monoidal
structures to be essentially the same.  In practice, I find that I
very rarely think about a braided monoidal category as if it were
equipped with two monoidal structures; rather I think of it as having
one monoidal structure together with an extra structure called a
"braiding."  So there are arguments on both sides of this issue, and
as John says probably neither usage will create any confusion.

Mike

On Mon, May 10, 2010 at 1:16 PM, John Baez <john.c.baez@gmail.com> wrote:
> Eduardo wrote:
>
>
>> Andre points out:
>>
>> "To call a monoidal category 1-braided is kind of confusing because there
>> is no commutation structure on a general monoidal category. A monoidal
>> category is 0-braided."
>>
>> Being an outsider, with no previous neither usage or opinion on this
>> terminology beyond just monoidal and/or tensor category, this seems to me
>> definitive, and more than enough to settle the question.
>
>
> I'm glad that's enough to convince you that Michael Batanin's terminology
> "monoidal = 1-braided" is inferior to Andre's "monoidal = 0-braided".
>
> But I think "braided = doubly monoidal" is even better.  After all, a
> monoidal category has one tensor product; a braided monoidal category has
> two compatible tensor products, and a symmetric monoidal category has three.
>
>
> But I will not lose sleep if Andre uses "k-braided" as a synonym for
> "(k+1)-tuply monoidal".  I don't see it causing any confusion. I just think
> it will create more +1's in various formulas.  E.g.: the classifying space
> of a k-braided n-category is a (k+1)-fold loop space.
>
> Best,
> jb
>


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Re: bilax_monoidal_functors

[Andre Joyal]

Dear Urs and John,

I see no real conflict between your terminology and mine.

I do use the notion of n-fold monoid in my work, for example
in my "Notes on Quasi-categories".
An *algebraic theory* is defined to be a (quasi-)category with finite products.
The n-fold tensor power of the theory of monoids M
is the theory of n-fold monoids = E(n)-monoids for every n.

I am sketching a proof of the Stabilisation Hypothesis at section 43.5 of my notes.
The hypothesis is formulated in terms of an equivalence of theories: 

<The theory of (n+2)-fold monoidal n-categories is equivalent 
to the theory of symmetric monoidal n-categories>.

It follows that the quasi-category of (n+2)-fold monoidal n-categories 
is equivalent to the quasi-category of symmetric monoidal n-categories.

Best, 
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Urs Schreiber
Date: lun. 10/05/2010 06:28
À: John Baez
Objet : categories: Re: bilax monoidal functors
 
> An n-tuply monoidal k-category is (conjecturally) a special sort of
> (n+k)-category

By the way, some progress on this is available from John Francis' and
Jacob Lurie's discussion of k-tuply monoidal (n,1)-categories as those
equipped with a little k-cubes action.

In particular there is a proof of the stabilization hypothesis for
(n,1)-categories this way, and an analog of the May recognition
theorem for parameterized oo-groupoids, i.e. (oo,1)-sheaves.

Some of this is summarized with pointers to references here:

   http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category

Best,
Urs

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Re: bilax_monoidal_functors?=

[Michael Batanin]

>> Andre points out:
>>
>> "To call a monoidal category 1-braided is kind of confusing because there
>> is no commutation structure on a general monoidal category. A monoidal
>> category is 0-braided."
>> Being an outsider, with no previous neither usage or opinion on this
>> terminology beyond just monoidal and/or tensor category, this seems to me
>> definitive, and more than enough to settle the question.

Well, I agree with Andre's argument but it does not convince me to use
Andre's terminology nor John's terminology (see my objections below).

The shift of numbers in Andre's terminmology is annoying when you try to
prove stabilisation hypothesis using higher braided operads. I hope to
talk about this proof in Genoa in a couple of months but it follows
readily from another atabilization theorem for n-braided operads. It is
   where I was more or less forced to call braided operads 2-braided
operads despite violation of ("foo" = "1-foo").

Another argument in favor of this terminology is that it provides a
uniform terminology in higher dimensions which agrees with E_n-algebra
point of view developed by Lurie and also his proof of stabilization
hypothesis (see Urs's message).

I agree that it creates some clash in low dimensions but I think it is
not a big deal since classical terminology does not have numbers (nobody
calls a monoidal category 0-braided or symmeteic monoidal category
2-braided monoidal). The low dimensional cases are important but they
are not always good models for higher dimension. As an example, -2 and
-1 categories as Baez and Dolan pointed out can be understood as one
pointed set and two pointed set correspondingly. Should we shift the
numbers and call category a 3-category?


> But I think "braided = doubly monoidal" is even better.  After all, a
> monoidal category has one tensor product; a braided monoidal category has
> two compatible tensor products, and a symmetric monoidal category has three.

The trouble is that n-monoidal categories already exist. They were
introduced my Balteanu, Fioderowicz, Shwantzl and Vogt. This is why I
also see n-tuply monoidal as confusing. I do not say that they sound
identical but certainly very close to each other.


> But I will not lose sleep if Andre uses "k-braided" as a synonym for
> "(k+1)-tuply monoidal".

I am glad to join John. I am also grateful to everybody participating
in this discussion. Terminology is a very important issue but I do not
think it is a crime to use a different one if the clarity of exposition
dictates it and if one acknowledges the existence of an alternative.  I
think I will continue to use my own  terminology but I am going to give
more explanation in the introduction   for those who like a different
one.

with best regards,
Michael.


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Re: bilax_monoidal_functors?=

[Toby Bartels]

Michael Batanin wrote in part:

>I agree that it creates some clash in low dimensions but I think it is
>not a big deal since classical terminology does not have numbers (nobody
>calls a monoidal category 0-braided or symmeteic monoidal category
>2-braided monoidal). The low dimensional cases are important but they
>are not always good models for higher dimension. As an example, -2 and
>-1 categories as Baez and Dolan pointed out can be understood as one
>pointed set and two pointed set correspondingly. Should we shift the
>numbers and call category a 3-category?

No, but it seems to me that you are doing something very much like this.

The concept of n-category makes sense for n as low as -2,
so it would be nice to renumber this so that we start at n = 0.
However, if we do so, then we need a word other than "-category";
if "category" = "3-category", then this violates "foo" = "1-foo".

Similarly, the concept of k-braided MC makes sense for k = -1,
so it would be nice to renumber this so that we start at k = 0.
However, if we do so, then we need a word other than "-braided MC";
if "braided MC" = "2-braided MC", then this violates "foo" = "1-foo".

So either we stick with Andre's numbering, inelegant as may be,
or we change Andre's "-braided MC" to John's "-tuply MC".
But you say, no, we do not need "foo" = "1-foo",
simply renumber so that "braided MC" = "2-braided MC".
That is like saying, renumber so that "category" = "3-category".
While it is a more elegant numbering, it is likely to be confusing.

I will say no more about it.  I will be happy to read your papers,
as long as you explain your terminology up front, as we all should.
I may grumble to myself at your violation of "foo" = "1-foo",
but I will nevertheless understand since you have explained.
(But if you later post to the categories list about it,
  then I may be confused if you don't recall the numbering.)


--Toby


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calculus, homotopy theory and more

[Andre Joyal]

Dear All,

The shift n-->n+1 which occurs in the terminologies

"n-braided monoidal category" = "(n+1)fold monoidal category"

"n-connected spaces" = "(n+1)fold loop spaces" 

is very natural. A similar shift occurs in calculus.
The analogy between calculus and homotopy theory is far reaching.
It is the basis of the theory of analytic functors of Goodwilie. 

http://www.math.brown.edu/faculty/goodwillie.html
http://arxiv.org/abs/math/0310481
http://ncatlab.org/nlab/show/Goodwillie+calculus

I would to describe the very elementary aspects of this theory.
I will also say a few things about the Breen-Baez-Dolan Stabilisation Hypothesis,
claiming that it is a theorem.

Let me denote by K[[x]] the ring of formal power series in one
variable over a field K. The ring K[[x]] bears some ressemblances
with the category of pointed homotopy types (= pointed spaces up 
to weak homotopy equivalences). The category of pointed 
homotopy types is a ring (the product is the smash product 
and the sum is the wedge).

K === the category of pointed sets 

K[[x]]=== the category of pointed homotopy types

x === the pointed circle.

The augmentation K[[x]]-->K 
=== the functor  pi_0: pointed homotopy types ---> pointed sets

The augmentation ideal J 
=== the subcategory of pointed connected spaces. 

The n+1 power of the augmentation ideal J^{n+1}
=== the subcategory of pointed n-connected spaces.

The product of an element in J^{n+1} with an element of J^{m+1}
is an element of J^{n+m+2} 
=== the smash product of a n-connected space with 
a m-connected space is (n+m+1)-connected.

Multiplication by x === the suspension functor.

Division by x === the loop space functor.
Notice here the difference: the loop functor is right adjoint to 
the suspension functor, not its inverse. Moreover,
the loop space of a space has a special structure (it is a group).
The ideal J=xK[[x]] is isomorphic to K[[x]] via division by x.
Similarly, the category of pointed connected spaces is equivalent to
the category of topological groups via the loop space functor
(it is actually an equivalence of model categories). 
More generally, the ideal J^{n+1} is isomorphic to K[[x]] via division by x^{n+1}.
Similarly, the category of n-connected space is equivalent to
the category of (n+1)-fold topological group (it is actually an
equivalence of model categories) via the (n+1)-fold loop space functor.



The quotient ring K[[x]]/J^{n+1} === the category of n-truncated homotopy types (=n-types)

The sequence of approximations of a formal power series f(x)=a_0+a_1x+...
a_0
a_0+a_1x
a_0+a_1x+a_2x^2
...
...

=== the Postnikov tower of a pointed homotopy type X: 
[pi0X]
[pi0X;pi1X]
[pi0X;pi1X,pi2X]
...
...
Here, pi0X is the set of components of X,
[pi0X;pi1X] is the fundamental groupoid of X,
[pi0X;pi1X,pi2X] is the fundamental 2-groupoid of X, etc.


The differences between f(x) and its successives approximations

R0 = f(x)-a_0               = a_1x+a_2x^2+a_3x^3+....
R1 = f(x)-(a_0+a_1x)        =      a_2x^2+a_3x^3+a_4x^4+....
R2 = f(x)-(a_0+a_1x+a_2x^2) =             a_3x^3+a_4x^4+a_5x^5+....

===the Whitehead tower of X,

C_0=[0;pi1X, pi2X, pi3X,....]
C_1=[0;0,pi2X,pi3X, pi4X,....]
C_2=[0;0,0,pi3X,pi4X,pi4X,....]
....
....

Here, C_0 is the connected component of X at the base point,
C_1 is the universal cover of X constructed by from paths starting at the base point,
C_2 is the universal 2-cover of X constructed from paths starting the base point, etc.


Division by x is shifting down the coefficients of a power series
If f(x)=a_1x+a_2x^2+..., then f(x)/x= a_1+a_1x^2+...
Similarly, the loop space functor is shifting down the homotopy groups
of a pointed space: if X=[a_0,a_1,a_2,...] then Loop(X)=[a_1,a_2,....].

Unfortunately, the suspension functor does not shift up the homotopy groups of a space.
It is however shifting the first 2n homotopy groups of n-connected space X (n geq 1)
by a theorem of Freudenthal:

http://en.wikipedia.org/wiki/Freudenthal_suspension_theorem
http://en.wikipedia.org/wiki/Hans_Freudenthal 
 
For example, if X=[0;0,a_2, a_3,...] then Susp(X)=[0;0,0,a_2,b_3...],
and if X=[0;0,0, a_3, a_4, a_5,...] then Susp(X)=[0;0,0, 0, a_3, a_4, b_5,...].
In other words, the canonical map 

X-->LoopSusp(X)

is a 2n-equivalence if X is n-connected (n geq 1). 
If X[2n] denotes the 2n-type of X (the 2n-truncation of X),
then we have a homotopy equivalence

X[2n]-->LoopSusp(X)[2n]=Loop(Susp(X)[2n+1]).

It follows that if X is a n-connected 2n homotopy type then
we have a homotopy equivalence

X--->Loop(X')

where X'=Susp(X)[2n+1]. The space X' is said to
be a *delooping* of X. By iterating this construction 
we can construct an infinite sequence of spaces

X=X_0, X_1, X_2,.... 

such that X_n=Loop(X_{n+1}). In other words,

*a n-connected 2n homotopy type is an infinite loop space (canonically)*

The (n+1)-fold loop space of a n-connected space 
is an E(n+1)-space (a E(n)-space is a model of the little n-cubes
operad of Boardman and Vogt, a E(1)-space is a monoid,
a E(2)-space is a braided monoid,...). 
The (n+1)-fold loop space functor induces an equivalence between the  
category of n-connected spaces and the category of group-like E(n+1)-space 
(a monoid M is said to be group-like if pi0(M) is a group).
Observe that the (n+1)-fold loop space of a 2n-type is a (n-1)-type.
Freudenthal theorem implies that

*If a (n-1) homotopy type has the structure of a group-like E(n+1)-space 
   then it has also the structure of an E(infty)-space (canonically)*

A nicer statement is obtained by shifting the index n by one.

* If a n-type has the structure of a group-like E(n+2)-space then it 
has also the structure of an E(infty)-space (canonically)*

The group-like condition can be dropped: 

*If a n-type has the structure of an E(n+2)-space then it has
the the structure of an E(infty)-space (canonically)*

This is a special case of the Stabilisation Hypothesis of Breen-Baez-Dolan;

*If a n-category has the structure of an E(n+2)-category then it has the structure of 
symmetric monoidal category (canonically)*

(Equivalently, *If a monoidal n-category is (n+1)-braided then it has the structure of 
symmetric monoidal category (canonically)*)

It is not difficult to verify that these statements are formally equivalent. 

The Breen-Baez-Dolan Stabilisation Hypothesis is a theorem.


Best, 
André


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Re: calculus, homotopy theory and more (corrected)

[Michael Batanin]

Dear Andre,

thank you for your very nice posting. If I understood correctly your 
proof of stabilization hypothesis it is based on classical Freudenthal
theorem. I can not resist sketching another proof (this is joint work 
with Clemens Berger and Denis-Charles Cisinski) from which Freudenthal 
theorem is a consequence.

It is based on the use of higher braided operads. Classically one can 
consider nonsymmetric, braided and symmetric operads with the values in 
a symmetric monoidal category V. If V is in addition a model category 
one can speak about contractible operads.  When V = Cat, for example,the 
category of  algebras of contractible nonsymmetric (braided, symmetric) 
operads is (=Quillen equivalent) to the category of monoidal categories 
(braided, symmetric). This prompts a line of thinking that the category 
   of  k-braided (weak) n-categories must  be an algebra of a 
contractible k-braided operad with values in Cat_n.

It is well known that we can not define k-braided operads in classical
terms (i.e. a sequence of objects with action of some groups  and a 
compatible substitution). The contradiction here is that for a 
contractible such operad to have a right homotopy type the  quotient of 
   its m-th space  by group actions must have the homotopy type of space 
of (nonordered) configuration of m-points in R^k. And this is not a 
K(\pi,1) space unless k=1,2,\infty.

Nevertheless, one can overcome this difficulty if instead of group 
action we consider an action of a category. Formally we consider a 
category of quasibijections of k-ordinals (or maps of pruned k-trees, 
which are bijections on the highest level) Q_k. The m-th connected 
component of the nerve of this category has homotopy type of the space 
of (nonordered) configuration of m-points in R^k. Then a k-collection is 
a contravaliant functor Q_k --> V. One can construct a category of 
operads  O_n(V) which have k-collections as underlying collections:

arXiv:0804.4165 Locally constant n-operads as higher braided operads. M. 
A. Batanin.

Finally, we define a k-braided operad to be an object of O_k(V) such 
that the action of any quasibijection is a weak equivalence. Obviously a 
contractible k-operad is a k-braided operad.

We want, actually, a bit more and define a model category of k-braided 
operads. This can be done by an appropriate Bousfield localization 
O_k^loc(V) of the model category O_k(V). The fibrant objects in 
O_k^loc(V) are precisely fibrant k-braided operads.

There is also a simple functor

    S: O_{k+1}(V) --> O_k(V)

induced by inclusion

s: Q_k --> Q_{k+1}

(add a tail to any pruned tree of height k).

This functor has a left adjoint, moreover, this pair of adjoint induces 
a pair of Quillen functors between localised categories
   S^{loc}: O_{k+1}^{loc}(V) <==> O_k^{loc}(V):F^{loc}

Theorem [Operadic stabilisation]:
Let V be n-truncated as a model category and k is greater or equal to n+2.
Then the Quillen adjunction
         S^{loc} |- F^{loc}
   is a Quillen equivalence. Moreover,
in this case the symmetrisation functor

sym:O_k^{loc}(V) --> SO(V)

is the left Quillen equivalence (SO(V) is the category of symmetric 
operads).

Proof. It follows from an explicit calculation of the derived left Kan 
extension along the functor s: Q_k --> Q_{k+1} and the fact that s 
induces an isomorphism of the m-th homotopy groups of the nerves when
m < k-2 , k>2 (k=2 is special and corresponds to the canonical 
homomorphism from  braid groups to symmetric groups).


Let us define the model category Br_k(V) of k-braided object in V as the 
category of one object, one arrow, ..., one (k-1)-arrow algebras of a 
contractible cofibrant k-operad. The (derived) Eckman-Hilton argument 
shows that it is Quilen equivalent to the category E_k(V) of 
E_k-algebras in V.


Let V be an n-truncated symmetric monoidal model category and k is 
greater or equal to n+2.
Then the total left derived functor  LF^{loc}   map contractible 
k-operads to contractible (k+1)-operads and Lsym map contractible 
operads to E_{\infty}-operads. So we have a Quillen equivalence of the 
categories

    E_k(V) --> Br_k(V) --> Br_{k+1}(Cat_n)--> E_{\infty}(V).

Corollary 1[BBD stabilization hypothesis]
Take V=Cat_n (for example, Cat_n = Rezk n-categories).

Corollary 2 [Freudenthal theorem]
   V = n-homotopy types.

I have little understanding of Goodwillie calculus but I know that 
operads play an important role in it.
It would be very interested to see what corresponds in calculus to 
operadic stabilization.

with best regards,
Michael.





Joyal wrote:
> Dear All,
> 
> The shift n-->n+1 which occurs in the terminologies
> 
> "n-braided monoidal category" = "(n+1)fold monoidal category"
> 
> "n-connected spaces" = "(n+1)fold loop spaces"
> 
> is very natural. A similar shift occurs in calculus.
> The analogy between calculus and homotopy theory is far reaching.
> It is the basis of the theory of analytic functors of Goodwilie.
> 
> http://www.math.brown.edu/faculty/goodwillie.html
> http://arxiv.org/abs/math/0310481
> http://ncatlab.org/nlab/show/Goodwillie+calculus
> 
> I would to describe the very elementary aspects of this theory.
> I will also say a few things about the Breen-Baez-Dolan Stabilisation 
> Hypothesis,
> claiming that it is a theorem.
> 
> Let me denote by K[[x]] the ring of formal power series in one
> variable over a field K. The ring K[[x]] bears some ressemblance
> with the category of pointed homotopy types (= pointed spaces up
> to weak homotopy equivalences). The category of pointed
> homotopy types is a ring (the product is the smash product
> and the sum is the wedge).
> 
> K === the category of pointed sets
> 
> K[[x]]=== the category of pointed homotopy types
> 
> x === the pointed circle.
> 
> The augmentation K[[x]]-->K
> === the functor  pi_0: pointed homotopy types ---> pointed sets
> 
> The augmentation ideal J
> === the subcategory of pointed connected spaces.
> 
> The n+1 power of the augmentation ideal J^{n+1}
> === the subcategory of pointed n-connected spaces.
> 
> The product of an element in J^{n+1} with an element of J^{m+1}
> is an element of J^{n+m+2}
> === the smash product of a n-connected space with
> a m-connected space is (n+m+1)-connected.
> 
> Multiplication by x === the suspension functor.
> 
> Division by x === the loop space functor.
> Notice here the difference: the loop functor is right adjoint to
> the suspension functor, not its inverse. Moreover,
> the loop space of a space has a special structure (it is a group).
> The ideal J=xK[[x]] is isomorphic to K[[x]] via division by x.
> Similarly, the category of pointed connected spaces is equivalent to
> the category of topological groups via the loop space functor
> (it is actually an equivalence of model categories).
> More generally, the ideal J^{n+1} is isomorphic to K[[x]] via division 
> by x^{n+1}.
> Similarly, the category of n-connected space is equivalent to
> the category of (n+1)-fold topological group (it is actually an
> equivalence of model categories) via the (n+1)-fold loop space functor.
> 
> 
> 
> The quotient ring K[[x]]/J^{n+1} === the category of n-truncated 
> homotopy types (=n-types)
> 
> The sequence of approximations of a formal power series f(x)=a_0+a_1x+...
> a_0
> a_0+a_1x
> a_0+a_1x+a_2x^2
> ...
> ...
> 
> === the Postnikov tower of a pointed homotopy type X:
> [pi0X]
> [pi0X;pi1X]
> [pi0X;pi1X,pi2X]
> ...
> ...
> Here, pi0X is the set of components of X,
> [pi0X;pi1X] is the fundamental groupoid of X,
> [pi0X;pi1X,pi2X] is the fundamental 2-groupoid of X, etc.
> 
> 
> The differences between f(x) and its successives approximations
> 
> R0 = f(x)-a_0               = a_1x+a_2x^2+a_3x^3+....
> R1 = f(x)-(a_0+a_1x)        =      a_2x^2+a_3x^3+a_4x^4+....
> R2 = f(x)-(a_0+a_1x+a_2x^2) =             a_3x^3+a_4x^4+a_5x^5+....
> 
> ===the Whitehead tower of X,
> 
> C_0=[0;pi1X, pi2X, pi3X,....]
> C_1=[0;0,pi2X,pi3X, pi4X,....]
> C_2=[0;0,0,pi3X,pi4X,pi4X,....]
> ....
> ....
> Here, C_0 is the connected component of X at the base point,
> C_1 is the universal cover of X constructed by from paths starting at 
> the base point,
> C_2 is the universal 2-cover of X constructed from paths starting the 
> base point, etc.
> 
> 
> Division by x is shifting down the coefficients of a power series
> If f(x)=a_1x+a_2x^2+..., then f(x)/x= a_1+a_2x+...
> Similarly, the loop space functor is shifting down the homotopy groups
> of a pointed space: if X=[a_0;a_1,a_2,...] then Loop(X)=[a_1;a_2,....].
> 
> Unfortunately, the suspension functor does not shift up the homotopy 
> groups of a space.
> It is however shifting the first 2n homotopy groups of n-connected space 
> X (n geq 1)
> by a theorem of Freudenthal:
> 
> http://en.wikipedia.org/wiki/Freudenthal_suspension_theorem
> http://en.wikipedia.org/wiki/Hans_Freudenthal
> 
> For example, if X=[0;0,a_2, a_3,...] (n=1) then Susp(X)=[0;0,0,a_2,b_3...],
> and if X=[0;0,0, a_3, a_4, a_5,...] (n=2) then Susp(X)=[0;0,0, 0,  a_3, 
> a_4, b_5,...].
> In other words, the canonical map
> 
> X-->LoopSusp(X)
> 
> is a 2n-equivalence if X is n-connected (n geq 1).
> If X[2n] denotes the 2n-type of X (the 2n-truncation of X),
> then we have a homotopy equivalence
> 
> X[2n]-->LoopSusp(X)[2n]=Loop(Susp(X)[2n+1]).
> 
> It follows that if X is a n-connected 2n homotopy type then
> we have a homotopy equivalence
> 
> X--->Loop(X')
> 
> where X'=Susp(X)[2n+1]. The space X' is said to
> be a *delooping* of X. By iterating this construction
> we can construct an infinite sequence of spaces
> 
> X=X_0, X_1, X_2,....
> 
> such that X_n=Loop(X_{n+1}). In other words,
> 
> *a n-connected 2n homotopy type is an infinite loop space (canonically)*
> 
> The (n+1)-fold loop space of a n-connected space
> is an E(n+1)-space (a E(n)-space is a model of the little n-cubes
> operad of Boardman and Vogt, a E(1)-space is a monoid,
> a E(2)-space is a braided monoid,...).
> The (n+1)-fold loop space functor induces an equivalence between the 
> category of n-connected spaces and the category of group-like E(n+1)-space
> (a monoid M is said to be group-like if pi0(M) is a group).
> Observe that the (n+1)-fold loop space of a 2n-type is a (n-1)-type.
> Freudenthal theorem implies that
> 
> *If a (n-1) homotopy type has the structure of a group-like E(n+1)-space
>   then it has also the structure of an E(infty)-space (canonically)*
> 
> A nicer statement is obtained by shifting the index n by one.
> 
> * If a n-type has the structure of a group-like E(n+2)-space then it
> has also the structure of an E(infty)-space (canonically)*
> 
> The group-like condition can be dropped:
> 
> *If a n-type has the structure of an E(n+2)-space then it has
> the the structure of an E(infty)-space (canonically)*
> 
> This is a special case of the Stabilisation Hypothesis of Breen-Baez-Dolan;
> 
> *If a n-category has the structure of an E(n+2)-category then it has the 
> structure of
> symmetric monoidal category (canonically)*
> 
> (Equivalently, *If a monoidal n-category is (n+1)-braided then it has 
> the structure of
> symmetric monoidal category (canonically)*)
> 
> It is not difficult to verify that these statements are formally equivalent.
> 
> The Breen-Baez-Dolan Stabilisation Hypothesis is a theorem.
> 
> 
> Best,
> André
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 


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Re: bilax_monoidal_functors

[Michael Shulman]

On Mon, May 10, 2010 at 2:28 PM, Jeff Egger <jeffegger@yahoo.ca> wrote:
> the fact that "autonomous category" is a special case (and, from one
> point of view, a rather uninteresting special case) of
> "star-autonomous category", whereas it sounds like "star-autonomous
> category" should mean an "autonomous category" with some extra
> structure.

I agree, it does sound like that, but there is at least a long
tradition of such names in mathematics (not that that makes
them a good thing).
(http://ncatlab.org/nlab/show/red+herring+principle)

One reason I like "autonomous" to mean a symmetric monoidal category
in which all objects have duals is that the only alternative names I
have heard for such a thing convey misleading intuition to me.  They
are sometimes called "compact closed" or (I think) "rigid" monoidal
categories, but "compact" and "rigid" are words with definite and
inapplicable intuitive meanings for me.  Compact means small, finite,
bounded, inaccessible by directed joins, etc. and "rigid" means "having few
automorphisms," and I don't see that there is anything very compact or
rigid about such categories.  The only relationship I can think of is that a
compact subset of a Hausdorff space is closed, and a symmetric monoidal
category with duals for objects is also automatically closed, but of course
these two meanings of "closed" are totally different.  Perhaps someone
can enlighten me?

Mike


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Re: calculus, homotopy theory and more (corrected)

[Michael Batanin]

Dear Andre,

> *If a n-type has the structure of an E(n+2)-space then it has
> the structure of an E(infty)-space (canonically)*
>
>
> which follows from the fact that the E(n+2) operad is n-connected.
> You may recall that we have discussed this in Barcelona.
> You told me that you knew that the E(n+2) operad is n-connected.
> I had learned it a week before from Lurie during my visit to Toen in Toulouse.

Yes, I remember this discussion. Actually my proof comes down to the
same fact since Q_n has homotopy type of unodered little cube
configurations in an n-cube. Lurie's proof is also based on the same fact.

Another approach to the proof that n-type with E_{n+2}-space structure
is also E_{infty}-space can be obtained by combining an idea of John
Baez and Jim Dolan of counting n-trees and my calculations of cells in
the Fulton-Macpherson operad. This is extremely simple combinatorial
proof. I can not reproduce it in this post because it requires some
pictures. But I remember, Andre, we discussed it with you in Montreal in
2004. I'll be happy to explain it again in Genoa.


> The rest of the proof of the Stabilisation hypothesis is formal but
> depends heavily on the machinery of (homotopical) universal algebra
> I have developed in my "Notes on quasi-categories".

The same for our proof. It does require a lot of homotopical algebra to
be able to localize model categories of operads.


> I believe that Goodwillie calculus is one of the next big thing in math.

I agree with it. It would be really wonderful if some experts organize a
   workshop on this subject with some introductory lectures for the
beginners.


>
> I look forward to see you in Genoa,

So do I.

Michael.




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Re: bilax_monoidal_functors?=

[Michael Batanin]

  >> Should we shift the
>> numbers and call category a 3-category?
>
> No, but it seems to me that you are doing something very much like this.

Not at all. It may be was not a good example. A better example would be
categories. If we follow the principle "foo = 1 foo" and want to agree
with historical low dimensional terminology we should call categories
2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.
There are many other examples like stack, gerbes and so on. I agree with
Mike Shulman that this is a byproduct of categorification. But we can
survive with it.

Concerning n-braided categories versus (n+1)-fold categories. Yes, I
would be happy to use (n+1)-fold terminology but it also clashes with
iterated monoidal categories of BFSW as I said.

Michael.


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Re: calculus, homotopy theory and more (corrected)

[Andre Joyal]

Dear Michael,

A basic ingredient in my approach to higher categories is the notion
of complete Segal space introduced by Rezk.
I have learned Rezk theory in proving the Quillen equivalence 
between quasi-categories and complete Segal spaces.
In my "Notes on Quasi-categories" I am introducing
an abstract notion of complete Segal space called
*Rezk category*, or *reduced category*.
A category object (internal to a quasi-category) is said 
to be *reduced* if its object of objects is 
*isomorphic* to its object of isomorphisms via the unit map. 
(an isomorphism in a quasi-category is an arrow which
is invertible in the homotopy category).
An ordinary category (in set) is reduced iff every
isomorphism is a unit, a very stringent condition.
Ordinary categories are seldom reduced (posets are).
Every reduced category is skeletal.
An equivalence between reduced categories is necessarly 
an isomorphism. In contrast, there are plenty of 
reduced categories in homotopy theory.
In fact every category internal to the quasi-category of spaces
is *equivalent*  to a reduced category (via a fully faith ess surj functor).
This key result was proved by Rezk for complete Segal spaces:
every Segal category is *equivalent* to a complete Segal space.
The theory of reduced categories is essentially (homotopy) algebraic 
(unlike ordinary category theory in which we need to expand the notion 
of isomorphism (of categories) with that of equivalence).

I do not have the time to explain more of the idea of my proof now. 
A sketch can be found in my "Notes on Quasi-categories".

You wrote:

>I can not reproduce it in this post because it requires some 
>pictures. But I remember, Andre, we discussed it with you in Montreal in
>2004. I'll be happy to explain it again in Genoa.
 
I hope I will understand this time!
I always find our conversation very stimulating!

See you in Genoa,

André


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Re: bilax_monoidal_functors

[Michael Shulman]

On Thu, May 13, 2010 at 6:09 PM, Michael Batanin <mbatanin@ics.mq.edu.au> wrote:
> Concerning n-braided categories versus (n+1)-fold categories. Yes, I
> would be happy to use (n+1)-fold terminology but it also clashes with
> iterated monoidal categories of BFSW as I said.

No one has suggested "(n+1)-fold monoidal" categories for that very
reason.  The terminology being suggested is "(n+1)-tuply monoidal."

Mike


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Re: terminology (was: bilax_monoidal_functors)

[Peter Selinger]

[Note from moderator: the correct sender, Peter Selinger, was 
inadvertently omitted from this post, sorry Peter!!]


Argh, Michael, you have managed to make a mess of the existing
terminology. The terminology is confusing, but it is actually
settled. While many concepts have more than one name, thankfully no
name refers to more than one concept so far (and I am working hard to
keep it that way - for example by discouraging redefinitions of
"autonomous"). Here are, for reference, the four most common notions
of (1-)categories with duals:

(1) An "autonomous category" is a monoidal category where every object
   has a left dual and a right dual. Note that it is not assumed to be
   symmetric. There is also the notion of a "left autonomous category",
   where only left duals are assumed, and analogously "right autonomous
   category". Note that duals, where they exist, are unique up to
   isomorphism, so being autonomous is a property of monoidal
   categories, not an additional structure.

   "Rigid category" is a synonym of "autonomous category", preferred by
   certain communities of authors.

(2) A "pivotal category" is an autonomous category equipped with a
   monoidal natural isomorphism A -> A**. (A right autonomous category
   with such an isomorphism is automatically left autonomous too, so
   the right/left distinction does not apply to pivotal categories).

   "Sovereign category" is a synonym of "pivotal category" used by
   Freyd and Yetter in one paper, but it does not seem to have caught
   on. It was a word play suggesting something that is even more than
   autonomous.

(3) A "tortile category" is a braided pivotal (equivalently balanced
   autonomous) category satisfying theta* = theta (where theta is the
   twist).

   "Ribbon category" is a synonym of "tortile category", preferred by
   certain communities of authors.

(4) A "compact closed category" is a tortile category that is symmetric
   (as a balanced monoidal category), or equivalently, an autonomous
   symmetric monoidal category.

Of course (4) => (3) => (2) => (1).  There are a number of in-between
concepts, which are generally less natural and of interest primarily
for technical reasons. Please see my recent survey "A survey of
graphical languages for monoidal categories" for a far more detailed
discussion (http://arxiv.org/abs/0908.3347). Particularly the table on
p.60 shows the whole taxonomy on one page.

I will briefly mention two of the "less natural" notions:

* A "braided autonomous" category is a monoidal category that is both
   braided and autonomous (with no axioms relating the two structures).
   This notion is entirely uninteresting, except to note that a braided
   left autonomous category is automatically right autonomous, due to
   the existence of isomorphisms A -> A**, and to note that it is NOT
   automatically pivotal, because said isomorphism is not monoidal.

* A "braided pivotal category" is a monoidal category that is both
   braided and autonomous (again with no axioms relating the two
   structures). This notion is also completely uninteresting, except to
   note that a braided pivotal category is exactly the same thing as a
   balanced autonomous category (because on a braided autonomous
   category, giving a pivotal structure is precisely equivalent to
   giving a balanced structure). Such categories were studied by Freyd
   and Yetter, but arguably they were superseded by the better notion
   of tortile categories. These categories have a graphical language up
   to "regular isotopy", which means that one of the three Reidemeister
   moves fails.

I have come to the opinion that it is a very good thing that notions
(1)-(3) above have distinct names, and are not just distinguished by
adjectives. It would be tempting to call a pivotal category a
"[something] autonomous category", and to call a tortile category for
example "[something else] braided pivotal" or "[something else]
balanced autonomous". But the most natural adjective for [something]
would be "pivotal", and the most natural adjective for [something
else] would be "tortile", which would only make the names longer
without adding any information.

I do believe that the term (4) "compact closed" is something of an
oddity, since "symmetric autonomous" would be similarly succinct, more
systematic, and much more descriptive - in fact, it requires no
additional definition if "symmetric" and "autonomous" have already
been defined. Also, as Michael has pointed out, the name "compact"
here has little to do with its usual meaning in mathematics.

If this concept were invented today, one should certainly call it
"symmetric autonomous". But in light of the fact that "compact closed"
was historially the first of notions (1)-(4) defined, and that the
term "compact closed" is already extremely well-known and wide-spread,
this is one case where I believe it is better to stick with the
existing terminology rather than trying to force it into a
taxonomy. That doesn't mean that slow incremental change is not
possible. For example, it seems reasonable to write "note that a
compact closed category is the same as a symmetric autonomous
category" whenever giving the definition. Perhaps after a few years,
people will write "note that a symmetric autonomous category is also
known as a compact closed category", and maybe after many more years,
the term "symmetric autonomous" will even become standard. But such
changes should come about through repeated and incremental use by a
community, and not by unilateral choices. As a general rule, I think
it is good manners when changing terminology (or inventing new
unsystematic terminology) to give the old (or systematic) terminology
in parentheses at least once per paper.

In Oxford, compact closed categories are nowadays called "compact
categories". I try not to follow this convention because it replaces
one bad term with a shorter, but equally bad one. It would also clash
with the standard meaning of "compact" in cases where the category was
actually a topological space. But it seems like a benign enough change
and is catching on rapidly.

My last comment is that, unlike what Jeff Egger claimed, "autonomous
category" is not a special case of "*-autonomous category", because no
symmetry is assumed in autonomous categories. Unless of course one
first drops symmetry from the definition of *-autonomous categories,
as Jeff has also suggested. As it stands, neither of "autonomous" and
"*-autonomous" implies the other, which is perfectly fine in my
opinion, since they are two different words.

-- Peter

Michael Shulman wrote:
>
> On Mon, May 10, 2010 at 2:28 PM, Jeff Egger <jeffegger@yahoo.ca> wrote:
>> the fact that "autonomous category" is a special case (and, from one
>> point of view, a rather uninteresting special case) of
>> "star-autonomous category", whereas it sounds like "star-autonomous
>> category" should mean an "autonomous category" with some extra
>> structure.
>
> I agree, it does sound like that, but there is at least a long
> tradition of such names in mathematics (not that that makes
> them a good thing).
> (http://ncatlab.org/nlab/show/red+herring+principle)
>
> One reason I like "autonomous" to mean a symmetric monoidal category
> in which all objects have duals is that the only alternative names I
> have heard for such a thing convey misleading intuition to me.  They
> are sometimes called "compact closed" or (I think) "rigid" monoidal
> categories, but "compact" and "rigid" are words with definite and
> inapplicable intuitive meanings for me.  Compact means small, finite,
> bounded, inaccessible by directed joins, etc. and "rigid" means "having few
> automorphisms," and I don't see that there is anything very compact or
> rigid about such categories.  The only relationship I can think of is that a
> compact subset of a Hausdorff space is closed, and a symmetric monoidal
> category with duals for objects is also automatically closed, but of course
> these two meanings of "closed" are totally different.  Perhaps someone
> can enlighten me?
>
> Mike
>



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Re: bilax_monoidal_functors?

[Toby Bartels]

Michael Batanin wrote:

>If we follow the principle "foo = 1 foo" and want to agree
>with historical low dimensional terminology we should call categories
>2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.

Sorry, but I don't think that you understand what we (Mike and I) mean
when we say that "foo" should equal "1-foo".

In all of these examples, the word "1-foo" (or "1-tuply foo")
means the same as the historic low-dimensional term "foo":
* "n-category", with the usual meaning;
* "n-set", as you suggested above;
* "k-tuply monoidal", as used by John Baez;
* "k-braided monoidal", as used by Andre Joyal;
* "n-stack", with the usual meaning;
* "n-sheaf", as Mike Shulman suggested.

In only these examples, the word "1-foo" does ~not~ mean the same as "foo":
* "n-connected space", with the usual meaning;
* "n-category", with the new meaning that you suggested earlier;
* "k-braided monoidal", as you used it here:
   http://permalink.gmane.org/gmane.science.mathematics.categories/5764/.

I like some of the terms in the first list more than others.
I find some of them sensibly numbered and some of them not
(which is part, but not all, of what goes into my liking them).
But I find all of them usable and I instantly understand them.

I object to the terms in the second list as inherently confusing,
even when I find them sensibly numbered.  Of the terms on that list,
only "n-connected space" has actually been sanctioned by history.
(But see http://ncatlab.org/nlab/show/k-simply+connected+n-category
for an alternative approach.)


--Toby


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Re: bilax_monoidal_functors

[Andre Joyal]

Dear Michael,

> Compact means small, finite,
> bounded, inaccessible by directed joins, etc. and "rigid" means "having few
> automorphisms," and I don't see that there is anything very compact or
> rigid about such categories.  The only relationship I can think of is that a
> compact subset of a Hausdorff space is closed, and a symmetric monoidal
> category with duals for objects is also automatically closed, but of course
> these two meanings of "closed" are totally different.  Perhaps someone
> can enlighten me?

 
I guess that in the category of R-modules over a commutative ring R, 
a module M has a (good) dual iff it is finitely generated projective
iff the endo-functor functor Hom(M,-) preserves all colimits
(M is *compact* in a strong sense). 

The rigidity terminology may have something to do with Tanaka duality.
If C is a rigid  monoidal category, then any monoidal natural 
transformation u:F-->G between (strong) monoidal functors C-->E
(where E is a monoidal category) is invertible.

I would prefer a different terminology for monoidal categories with duals.

What about "auto-dual monoidal category"?

It as a bit like "autonomous" category.

Best,
André





 



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Re: terminology (was: bilax_monoidal_functors)

[Michael Shulman]

I apologize for making a "mess" by including symmetry where it was not
supposed to be; I thought I was just repeating what I had heard
elsewhere.  Perhaps my memory was faulty, or perhaps someone else made
the same error (or mess).  Jeff's comment probably contributed to my
confusion too.  (I have read your very nice paper, but since I am only
interested in a few of the notions, I didn't take the trouble to
memorize all the different shadings of meaning.)  Thanks for setting
me straight.

I am happy with "symmetric autonomous," although I guess it loses out
to "compact" on the score of brevity.

Mike

On Fri, May 14, 2010 at 9:43 AM, Peter Selinger
<selinger@mathstat.dal.ca> wrote:
> Argh, Michael, you have managed to make a mess of the existing
> terminology. The terminology is confusing, but it is actually
> settled. While many concepts have more than one name, thankfully no
> name refers to more than one concept so far (and I am working hard to
> keep it that way - for example by discouraging redefinitions of
> "autonomous"). Here are, for reference, the four most common notions
> of (1-)categories with duals:
>
> (1) An "autonomous category" is a monoidal category where every object
>  has a left dual and a right dual. Note that it is not assumed to be
>  symmetric. There is also the notion of a "left autonomous category",
>  where only left duals are assumed, and analogously "right autonomous
>  category". Note that duals, where they exist, are unique up to
>  isomorphism, so being autonomous is a property of monoidal
>  categories, not an additional structure.
>
>  "Rigid category" is a synonym of "autonomous category", preferred by
>  certain communities of authors.
>
> (2) A "pivotal category" is an autonomous category equipped with a
>  monoidal natural isomorphism A -> A**. (A right autonomous category
>  with such an isomorphism is automatically left autonomous too, so
>  the right/left distinction does not apply to pivotal categories).
>
>  "Sovereign category" is a synonym of "pivotal category" used by
>  Freyd and Yetter in one paper, but it does not seem to have caught
>  on. It was a word play suggesting something that is even more than
>  autonomous.
>
> (3) A "tortile category" is a braided pivotal (equivalently balanced
>  autonomous) category satisfying theta* = theta (where theta is the
>  twist).
>
>  "Ribbon category" is a synonym of "tortile category", preferred by
>  certain communities of authors.
>
> (4) A "compact closed category" is a tortile category that is symmetric
>  (as a balanced monoidal category), or equivalently, an autonomous
>  symmetric monoidal category.
>

...

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terminology

[Joyal, André]

Micheal Batanin wrote

>If we follow the principle "foo = 1 foo" and want to agree
>with historical low dimensional terminology we should call categories
>2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.

A few thoughts about terminology.

Categories are tradidionally named according to the nature
of their objects, not the nature of their morphisms. 
We say "the category of sets" not "the category of functions". 
This convention is not respected in the case where the category 
has only one object: we call it a monoid, not because it is a 
mono-object category (maybe we should) but because it has only one 
binary operation in contrast with a ring. 
Like monoids, operads are collections of abstract operations
closed under composition. Classical operads have
only one object, one color. But multi-colored operads
are often called muti-categories, especially when they are big. 

A set is a discrete homotopy type, a 0-type.
This why I like to give the category of sets rank 0.
I like to denote the quasi-category of n-types by U[n].


Best,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Michael Batanin
Date: jeu. 13/05/2010 19:09
À: Toby Bartels
Objet : categories: Re: bilax_monoidal_functors?=
 

   >> Should we shift the
>> numbers and call category a 3-category?
>
> No, but it seems to me that you are doing something very much like this.

Not at all. It may be was not a good example. A better example would be
categories. If we follow the principle "foo = 1 foo" and want to agree
with historical low dimensional terminology we should call categories
2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.
There are many other examples like stack, gerbes and so on. I agree with
Mike Shulman that this is a byproduct of categorification. But we can
survive with it.


...

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Re: bilax_monoidal_functors

[Jeff Egger]

>> But I think "braided = doubly monoidal" is even
> > better.  After all, a
> > monoidal category has one tensor product; a braided
> > monoidal category has
> > two compatible tensor products, and a symmetric
> > monoidal category has three.
> 
> The trouble is that n-monoidal  categories already exist.
> They were
> introduced my Balteanu, Fioderowicz, Shwantzl and Vogt.
> This is why I
> also see n-tuply monoidal as confusing. I do not say that
> they sound
> identical but certainly very close to each other.

This is a strong point.  Obviously n-tuply monoidal category
should mean category with n "compatible" monoidal structures;
but there many possible meanings of "compatible".  One choice 
leads to a single monoidal structure with an (n-1)-braiding;
but a different  choice leads to the notion of BFSV.  In fact,
I think that even the BFSV  notion is too strict---it forces 
all the units to be the same, where I think one should allow
them to be different (in general).  That is, I think it would
be reasonable to use "doubly monoidal category" to mean 
(pseudo)monoid internal to LAX (rather than STRONG, or even 
NORMAL).  

Cheers,
Jeff.





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Re: terminology

[Toby Bartels]

Thanks for this list, Peter!

I have put much of its content on the nLab at
http://ncatlab.org/nlab/show/category+with+duals
(and Mike has already put more on pages linked from there),
so feel free to speak up again (here or by editing those pages)
if something is wrong.


--Toby


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Re: Terminology

[Jean Benabou]

Dear Steve,

I totally agree with you.
Let me apply  your zoological criteria to Category Theory. You begin  
with the very simple notions of category, functor and natural   
transformation. But then you start piling in subclauses such as  
categories with finite limits, or regular, or abelian, or the glorious  
toposes. For functors you refine the notion to fully faithful ones or  
those who have an adjoint, or are flat, or are fibrations. I could  
give hundreds of examples, and even a meticulous zoologist would  
say:To much is to much!
Obviously Category Theory is very bad and the very idea of putting in   
a same bag groups, topological spaces, locales, and the glorious  
toposes is a misconception.
Serious mathematicians agreed with this. You are too young to remember  
the time when these mathematicians called, with zoological  
justification, this theory :  Abstract general nonsense.

All the best,

Jean




Le 14 févr. 17 à 09:48, Steve Vickers a écrit :

> Dear Fred,
>
> A good answer, but my point was that it was a bad question.
>
> You see this once you start pressing at the details. Are seals and  
> turtles fish? No, but on your definition it depends on whether  
> flippers count as legs or not. What about sea snakes? Obviously not  
> - they're snakes, that just happen to live in the sea. But then eels  
> do seem a bit more fishy.
>
> A meticulous zoologist would start piling on the subclauses to pin  
> it down more precisely, but we know that that does not actually  
> refine our understanding of zoology. It just amplifies the  
> misconceptions underlying the original question.
>
> I'm saying the same can happen in mathematics.
>
> All the best,
>
> Steve.
>
>> On 11 Feb 2017, at 20:42, Fred E.J. Linton <fejlinton@usa.net> wrote:
>>
>> Steve, et al.,
>>
>> If you want
>>
>>> a definition of "fish", but on the understanding that it has to  
>>> include
>> whales
>>
>> let me offer: "legless marine vertebrates" :-) .
>>
>> Cheers, -- tlvp
>>
>



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Re: Terminology

[Steve Vickers]

Dear Fred,

A good answer, but my point was that it was a bad question.

You see this once you start pressing at the details. Are seals and turtles fish? No, but on your definition it depends on whether flippers count as legs  or not. What about sea snakes? Obviously not - they're snakes, that just happen to live in the sea. But then eels do seem a bit more fishy.

A meticulous zoologist would start piling on the subclauses to pin it down more precisely, but we know that that does not actually refine our understanding of zoology. It just amplifies the misconceptions underlying the original  question.

I'm saying the same can happen in mathematics.

All the best,

Steve.

> On 11 Feb 2017, at 20:42, Fred E.J. Linton <fejlinton@usa.net> wrote:
> 
> Steve, et al.,
> 
> If you want 
> 
>> a definition of "fish", but on the understanding that it has to include
> whales
> 
> let me offer: "legless marine vertebrates" :-) .
> 
> Cheers, -- tlvp
> 



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Re: Terminology

[Fred E.J. Linton]

Steve, et al.,

If you want 

> a definition of "fish", but on the understanding that it has to include
whales

let me offer: "legless marine vertebrates" :-) .

Cheers, -- tlvp



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Re: Terminology

[Steve Vickers]

Dear Jean,

My own understanding (superficial and possibly wrong) of the history is that  since Bourbaki there have been definitions of "structure" with the aim of reconciling the algebraic examples (where the homomorphisms preserve structure) with the topological spaces (where the continuous maps have inverse images that preserve structure). Certainly if you look at Joy of Cats, the prime classes of examples are those of topological and algebraic categories.

But, as we know from topos theory, it is not foundationally robust to treat topological spaces as "sets with structure", i.e. point-set topology. In general we have to work point-free, at least if we want to save important parts of topology from going down the drain.

If such an important source of examples, the point-set topological spaces, turned out to be misleading, then, in retrospect, any "precise meaning [of structure] on which the community of mathematicians agree", was probably misguided.

It's like looking for a definition of "fish", but on the understanding that it has to include whales.

All the best,

Steve.

> On 9 Feb 2017, at 16:38, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
> 
> Dear Christopher,
> What I, personally, mean by structure is not the point. This word is used,  very often, in mathematical texts. Sometimes giving the impression that it has a precise meaning on which the community of mathematicians agree. And I was sure there was at least one definition on which the majority of users did  agree
> 
> Then I received 3 answers all referring to: The joy of Cats, but different:
> For Carsten Führman, only faithfulness is required, which obviously is not enough
> Jiri Adamek adds: an isomorphism in S is an identity if its image is. I agree with this; but again not enough.
> Thomas Streicher adds a third condition, with which I would probably agree  if was sure of the precise meaning of isofibration. Could you please, even at the risk of being pedantic say what you mean by that
> 
> Many thanks to all
> 
> 
>> â€ژHi Jean - I don't quite understand this question but  would like to. What do you mean by 'structure'? Thanks
>> 

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Re: Terminology

[George Janelidze]

Dear Jean and Jiri,

As we know there is no such notion accepted by everybody. I would probably
vote for

faithful + amnestic + iso-fibration.

Best regards, George

--------------------------------------------------
From: "Jir? Ad?mek" <j.adamek@tu-bs.de>
Sent: Wednesday, February 8, 2017 6:34 PM
To: "categories net" <categories@mta.ca>
Subject: categories: Re: Terminology

> Dear Jean,
>
> The simplest answer is: faithful. But a better one (in view of
> `everything up to isomorphism') is: faithful and amnestic. The latter
> means that p reflects identity morphisms: an isomorphism in S is an
> identity if its image by p is. See The Joy of Cats (free on the web).
>
> Best, Jiri
>
>> QUESTION
>> Let  p: S --> X  be a functor. What conditions should satisfy p to be
>> called a structure functor, i.e. such that every object  s of S can be
>> thought of as a structure on the object  p(s).
>


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Terminology

[Andrée Ehresmann]

For Charles Ehresmann, the answer to Jean's question was that p be a
"homomorphism functor", a notion he already defined in his 1957 paper
"Gattungen in Lokalen Strukturen", reprinted in

   http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_I-1_et_I-2.pdf

In modern terms it should correspond to a faithful and amnestic functor.

Cordially
Andree



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Re: Terminology

[Jean Benabou]

Dear Christopher,
What I, personally, mean by structure is not the point. This word is  
used, very often, in mathematical texts. Sometimes giving the  
impression that it has a precise meaning on which the community of  
mathematicians agree. And I was sure there was at least one definition  
on which the majority of users did agree

Then I received 3 answers all referring to: The joy of Cats, but  
different:
For Carsten Führman, only faithfulness is required, which obviously is  
not enough
Jiri Adamek adds: an isomorphism in S is an identity if its image is.  
I agree with this; but again not enough.
Thomas Streicher adds a third condition, with which I would probably  
agree if was sure of the precise meaning of isofibration. Could you  
please, even at the risk of being pedantic say what you mean by that

Many thanks to all


> â€ژHi Jean - I don't quite understand this question but would  
> like to. What do you mean by 'structure'? Thanks
>
> Sent from my BlackBerry 10 smartphone on the O2 network.
>  Original Message
> From: Jean Benabou
> Sent: Wednesday, 8 February 2017 16:18
> To: Categories
> Reply To: Jean Benabou
> Subject: categories: Terminology
>
>
> Dear all,
>
> I'm sure the following question has been answered to. Could anyone
> give me a precise answer and references to this answer. Many thanks.
>
> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).
>

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Re: Terminology

[Thomas Streicher]

Dear Jean,

in Remark 13.18 of their book on "Algebraic Theories" Adamek, Rosicky
and Vitale suggest the following conditions

1) p faithful (what they call "concrete over X")
2) p-vertical isos are identities (what they call "amnestic"))
3) p is an isofibration (what they call "transportable")

These seem to be reasonable conditions validated by most examples.

Does this confirm with your intuition?

Thomas

> I'm sure the following question has been answered to. Could anyone
> give me a precise answer and references to this answer. Many thanks.
>
> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).


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Re: Terminology

[Carsten Führmann]

Dear Jean,

unless there is a technical meaning of "structure" I'm not aware of, the
answer may be "Concrete categories" in the sense of Adámek, Herrlich, and
Strecker: http://katmat.math.uni-bremen.de/acc/acc.pdf. A concrete category
is just a faithful functor, but a remarkable amount of theory can be build
on that notion. In particular, a classification of "algebra-like" and
"space-like" structures is already possible at that level.



On Wed, Feb 8, 2017 at 4:56 PM Jean Benabou <jean.benabou@wanadoo.fr> wrote:

> Dear all,
>
> I'm sure the following question has been answered to. Could anyone
> give me a precise answer and references to this answer. Many thanks.
>
> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).
>

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Re: Terminology

[Jirí Adámek]

Dear Jean,

The simplest answer is: faithful. But a better one (in view of
`everything up to isomorphism') is: faithful and amnestic. The latter
means that p reflects identity morphisms: an isomorphism in S is an
identity if its image by p is. See The Joy of Cats (free on the web).

Best, Jiri

> QUESTION
> Let  p: S --> X  be a functor. What conditions should satisfy p to be
> called a structure functor, i.e. such that every object  s of S can be
> thought of as a structure on the object  p(s).


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Terminology

[Jean Benabou]

Dear all,

I'm sure the following question has been answered to. Could anyone
give me a precise answer and references to this answer. Many thanks.

QUESTION
Let  p: S --> X  be a functor. What conditions should satisfy p to be
called a structure functor, i.e. such that every object  s of S can be
thought of as a structure on the object  p(s).



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Re: Terminology

[Robert Dawson]

On 02/05/2013 12:57 AM, Fred E.J. Linton wrote:
> Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:
>
>> ... I'd call it "essentially subterminal".
>
> Hmm ... hitting a translation engine in a particularly good mood, I found
> "essentially terminal" rendering, in German, as "wesentlich unheilbar".
>
> (Round-tripping from there, you get "fundamentally incurable". Like that?
> Alas, it drew a blank on the actual proposal, "essentially subterminal" :-)

 	Subterminal? Um, that would be "Unterseebootendbahnhof"?

 	<grin, duck, & run>



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Re: Terminology

[Fred E.J. Linton]

Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:

> ... I'd call it "essentially subterminal".

Hmm ... hitting a translation engine in a particularly good mood, I found  
"essentially terminal" rendering, in German, as "wesentlich unheilbar".

(Round-tripping from there, you get "fundamentally incurable". Like that?  
Alas, it drew a blank on the actual proposal, "essentially subterminal" :-)
.)

Cheers, -- Fred



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Re: Terminology

[Fred E.J. Linton]

Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:

> ... I'd call it "essentially subterminal".

Hmm ... hitting a translation engine in a particularly good mood, I found  
"essentially terminal" rendering, in German, as "wesentlich unheilbar".

(Round-tripping from there, you get "fundamentally incurable". Like that?  
Alas, it drew a blank on the actual proposal, "essentially subterminal" :-)
.)

Cheers, -- Fred



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Re: Terminology

[Fred E.J. Linton]

Forgive my repeating, perhaps unnecessarily, the obvious, but
without doing that I fear I may just get inextricably lost as I
try, once again, to sort my way through this question to more of
an answer than I was able to access the last times I tried.

If we pay momentary attention to the "underlying point-set" functor,
from the category of Topological Spaces to that of Sets, we see that
it "has" both a left adjoint, assigning to each set that self-same set
in its discrete topology, and a right adjoint, assigning to each set
that self-same set in its indiscrete topology.

That said, let me turn instead to the "underlying set of objects"
functor from that category of all small categories to that of sets.
It, too, has both a left adjoint, assigning to each set "the" category
having that self-same set as its set of objects, but admitting no 
morphisms between any two objects other than identity maps where 
identity maps are absolutely required -- what's known as the discrete
category on that set of objects -- and a right adjoint, assigning to
each set "the" category having that self-same set as its set of objects,
with the peculiar feature that each of its hom-sets has cardinality 1
-- category that, by analogy with the topological right adjoint, one
might (as Toby Bartels so deftly reminds us) choose to call indiscrete.

And if these categories are nothing more nor less than those that Jean 
Bénabou envisages, with functor to the terminal category 1 fully faithful,
then I guess "indiscrete" would be my answer, too, to his question,
"what would you call" such a category? But for me the indiscreteness
is not in any way a reflection of that full fidelity -- rather, it is
a reflection of the parallel between the fact that such a category "is" 
the value of the right adjoint to the "underlying set of objects" functor
and that an indiscrete space serves as value of the right adjoint to the 
"underlying point-set" functor.

"Setoïd"? "essentially subterminal"? Come on, folks, give us a break :-) !

Cheers, -- Fred

------ Original Message ------
Received: Mon, 29 Apr 2013 07:53:37 PM EDT
From: Toby Bartels <categories@TobyBartels.name>
To: Categories <categories@mta.ca>
Subject: categories: Re: Terminology

> Thomas Streicher wrote:
> 
>>Jean Bénabou wrote:
> 
>>>What would you call a category X such that the functor X --> 1 is
>>>full and faithful? Please don't tell me what they are, I  know that.
> 
>>Sticking to the pattern I suggested I'd call it "essentially subterminal".
> 
> I learnt to call that an "indiscrete category", so I probably would.
> (Another term that I've heard is "chaotic category", which I never liked.)
> Of course, I could also call it a "truth value",
> but only in a context where I would expect this to be understood
> (and being "non-evil", that is working up to equivalence,
> is not actually sufficient for that).  Thus the nLab has
> http://ncatlab.org/nlab/show/indiscrete+category as its own page.
> 
>>>Non evil is essentially evil.
>>>I rather like this conclusion, don't you?
> 
> It is beautiful, but is it accurate?
> 
>>I'd expect the people abhoring evilness would
>>say that full and faithful and essentially surjective is an "evil" notion
>>of equivalence as opposed to the "good" one of adjoint pair where unit  and
>>counit are isos. The latter makes sense in any 2-category whereas the
former
>>doesn't. However, often you just get the "evil" version when not having
>>a strong form of AC (for classes) available.
> 
> On the contrary, an ff and eso functor between two categories
> is enough for the people who abhor evil, as far as I know,
> to decide that the categories are equivalent (and so essentially the same).
> Yet at the same time, these people tend to abhor AC!  How can this be?
> It works if one works in a 2-category whose 1-morphisms are anafunctors.
> Then it is a theorem requiring no choice (and true internal to any topos)
> that any ff and eso functor can be enriched to an adjoint equivalence
> (and in an essentially unique way).
> 
> Of course, "abhor" here should really be read as "consider optional".
> It is possible to work with strict categories, or to work with AC,
> but the main principles and results of category theory do not require
either.
> 
> 
> --Toby
> 


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Re: Terminology

[Peter May]

I apologize for poor taste, but I do like chaotic: one might substitute
indiscrete
in the title ``Chaotic categories and equivariant classifying spaces''
(posted at
http://front.math.ucdavis.edu/1201.5178), but surely not essentially
subterminal.
The comment I'd really like to make is that such categories can be
seriously
interesting.

Peter


On 4/29/13 3:05 PM, Toby Bartels wrote:
> Thomas Streicher wrote:
>
>> Jean B?nabou wrote:
>>> What would you call a category X such that the functor X --> 1 is
>>> full and faithful? Please don't tell me what they are, I  know that.
>> Sticking to the pattern I suggested I'd call it "essentially subterminal".
> I learnt to call that an "indiscrete category", so I probably would.
> (Another term that I've heard is "chaotic category", which I never liked.)
> Of course, I could also call it a "truth value",
> but only in a context where I would expect this to be understood
> (and being "non-evil", that is working up to equivalence,
> is not actually sufficient for that).  Thus the nLab has
> http://ncatlab.org/nlab/show/indiscrete+category as its own page.
>
>>> Non evil is essentially evil.
>>> I rather like this conclusion, don't you?
> It is beautiful, but is it accurate?
>
>> I'd expect the people abhoring evilness would
>> say that full and faithful and essentially surjective is an "evil" notion
>> of equivalence as opposed to the "good" one of adjoint pair where unit and
>> counit are isos. The latter makes sense in any 2-category whereas the former
>> doesn't. However, often you just get the "evil" version when not having
>> a strong form of AC (for classes) available.
> On the contrary, an ff and eso functor between two categories
> is enough for the people who abhor evil, as far as I know,
> to decide that the categories are equivalent (and so essentially the same).
> Yet at the same time, these people tend to abhor AC!  How can this be?
> It works if one works in a 2-category whose 1-morphisms are anafunctors.
> Then it is a theorem requiring no choice (and true internal to any topos)
> that any ff and eso functor can be enriched to an adjoint equivalence
> (and in an essentially unique way).
>
> Of course, "abhor" here should really be read as "consider optional".
> It is possible to work with strict categories, or to work with AC,
> but the main principles and results of category theory do not require either.
>
>
> --Toby


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Re: Terminology

[Toby Bartels]

Thomas Streicher wrote:

>Jean B?nabou wrote:

>>What would you call a category X such that the functor X --> 1 is
>>full and faithful? Please don't tell me what they are, I  know that.

>Sticking to the pattern I suggested I'd call it "essentially subterminal".

I learnt to call that an "indiscrete category", so I probably would.
(Another term that I've heard is "chaotic category", which I never liked.)
Of course, I could also call it a "truth value",
but only in a context where I would expect this to be understood
(and being "non-evil", that is working up to equivalence,
is not actually sufficient for that).  Thus the nLab has
http://ncatlab.org/nlab/show/indiscrete+category as its own page.

>>Non evil is essentially evil.
>>I rather like this conclusion, don't you?

It is beautiful, but is it accurate?

>I'd expect the people abhoring evilness would
>say that full and faithful and essentially surjective is an "evil" notion
>of equivalence as opposed to the "good" one of adjoint pair where unit and
>counit are isos. The latter makes sense in any 2-category whereas the former
>doesn't. However, often you just get the "evil" version when not having
>a strong form of AC (for classes) available.

On the contrary, an ff and eso functor between two categories
is enough for the people who abhor evil, as far as I know,
to decide that the categories are equivalent (and so essentially the same).
Yet at the same time, these people tend to abhor AC!  How can this be?
It works if one works in a 2-category whose 1-morphisms are anafunctors.
Then it is a theorem requiring no choice (and true internal to any topos)
that any ff and eso functor can be enriched to an adjoint equivalence
(and in an essentially unique way).

Of course, "abhor" here should really be read as "consider optional".
It is possible to work with strict categories, or to work with AC,
but the main principles and results of category theory do not require either.


--Toby


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Re: Terminology

[Olivier Gerard]

On Sun, Apr 28, 2013 at 5:49 AM, Jean Bénabou <jean.benabou@wanadoo.fr>
  wrote:

I don't like very much "setoids", and I am very tempted by "essentially
> discrete" as Thomas suggested.


For these ones, I would suggest "catégories timides" or "catégories
réservées", as a play on "discrete", something you could translate as "shy"
or "bashful" or "shrinking categories".

What would you call a category X such that the functor X --> 1 is full and
> faithful? Please don't tell me what they are, I  know that. I'm not even
> asking if there is a we'll established name for them. I don't think there
> is one. What I ask is: Could you suggest one? Preferably a name which would
> be suitable when we work with categories internal to a Topos E where
> supports don't split.


If you are in a playful mood, one could call them "catégories unspirées".
Another suggestion is "catégories modestes".  This would make a good trio
with "catégories discrètes".

Olivier Gérard

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Re: Terminology

[Thomas Streicher]

Dear Jean,

> What would you call a category X such that the functor X --> 1 is
> full and faithful? Please don't tell me what they are, I  know that.

Sticking to the pattern I suggested I'd call it "essentially subterminal".

> Non evil is essentially evil.
> I rather like this conclusion, don't you?

Of course, that's brilliant dialectics! I'd expect the people abhoring evilness
would say that full and faithful and essentially surjective is an "evil" notion
of equivalence as opposed to the "good" one of adjoint pair where unit and
counit are isos. The latter makes sense in any 2-category whereas the former
doesn't. However, often you just get the "evil" version when not having
a strong form of AC (for classes) available. That's why your dialectics
definitely applies!

Best regards, Thomas



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Re: Terminology

[Jean Bénabou]

Dear Thomas, Dear all,

My definition of poset is: "preordered set". I don't know if there is a general agreement, since some answers seemed to suppose that I meant "partially ordered set". It is because I feared this confusion that i specified by adding: equivalent to a discrete category. 
Of course I knew that they were "equivalence relations", and had also many other simple characterizations. One which I like and need is: X is equivalent to a discrete category iff the functor X --> 1 is faithful  and conservative (i.e. reflects isos) because it has the following generalization:

Let P: X --> S be a prefibration. The following are equivalent:
(i) P is equivalent to a discrete fibration.
(ii) P is faithful and consevative.
(iii) each fiber of P is equivalent to a discrete category.

Thus my question  was not:  what are such categories, for which I knew perfectly many answers, but : is there a well established name for them? 
Suggestions such as "setoids" or "essentially discrete" show that this is not the case.

I don't like very much "setoids", and I am very tempted by "essentially discrete" as Thomas suggested. 
But I shall make my question a bit more difficult. 
What would you call a category X such that the functor X --> 1 is full and faithful? Please don't tell me what they are, I  know that. I'm not even asking if there is a we'll established name for them. I don't think there is one. What I ask is: Could you suggest one? Preferably a name which would be suitable when we work with categories internal to a Topos E where supports don't split.

As a side remark, let me say that I don't care very much for the distinction between "evil" and "non evil". Apart  from obvious moral or philosophical reasons, for the following purely mathematical one: Non-evilness depends on the notion of equivalence of categories. And this in turn may heavily depend on which notion of equivalence you chose. And some of these notions depend on the axiom of choice, which I might be tempted to call "evil". Thus we'd reach the following conclusion: 
Non evil is essentially evil. 
I rather like this conclusion, don't you?

Best regards,
Jean

Le 27 avr. 2013 à 15:08, Thomas Streicher a écrit :

> Dear Jean,
> 
a
> 
>> As many of you I presume, I have for ages, and very often, had to deal with categories which are both groupoïds and posets, or again which are equivalent to a discrete category. Is there a well established name for them?
> 
> What about "essentially discrete" like in "essentially small" or
> "essentially surjective". Generally, for any property P of categories
> I would say a category is "essentially P" if it is equivalent to a
> category with property P.
> So "essentially" is a kind of magic word transforming "evil" properties
> into "non-evil" ones. (I don't think one should always do this!)
> 
> Thomas



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Re: Terminology

[Thomas Streicher]

Dear Jean,

> As many of you I presume, I have for ages, and very often, had to deal with categories which are both groupo?ds and posets, or again which are equivalent to a discrete category. Is there a well established name for them?

What about "essentially discrete" like in "essentially small" or
"essentially surjective". Generally, for any property P of categories
I would say a category is "essentially P" if it is equivalent to a
category with property P.
So "essentially" is a kind of magic word transforming "evil" properties
into "non-evil" ones. (I don't think one should always do this!)

Thomas


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Re: Terminology

[David Roberts]

One option is "setoid".

Best regards,
David Roberts
On Apr 25, 2013 7:38 AM, "Jean Bénabou" <jean.benabou@wanadoo.fr> wrote:

> Dear all,
>
> As many of you I presume, I have for ages, and very often, had to deal
> with categories which are both groupoïds and posets, or again which are
> equivalent to a discrete category. Is there a well established name for
> them?
>
> Best regards,
> Jean

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Terminology

[Jean Bénabou]

Dear all,

As many of you I presume, I have for ages, and very often, had to deal with categories which are both groupoïds and posets, or again which are equivalent to a discrete category. Is there a well established name for them?

Best regards,
Jean

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Re: terminology

[Todd Trimble]

In reference to Eduardo's recent comment

>I feel the need to clarify some of my postings.
>
> Due to some public and private mails I realized that most people though
> that I
> was talking about the nLab.
> Well, all the time, when referring to "ghetto" or "subculture" I was
> aiming to
> the WHOLE of the category community within mathematics, not at the nLab
> within
> the category community.
>

I for one didn't get the impression you were referring to the
nLab.  My own comment was in response to Andre Joyal who
wrote "The 'evil' terminology is promoted by a small group of
peoples active in the nLab. It does not reflect a commun usage
in themathematical community."

I thought this could lead to a misunderstanding about the nLab,
hence my comment.

> Actually, I was not even aware of the existence of the nLab.
Due to this
> controversy, I visit the nLab and at first sight I essentially (not fully)
> agree with Andre's comments about the nLab in his recent posting.
>
> I say, go ahead !, nice work !
>
> I can add that I liked the lack of solemnity and the freedom to write down
> your understanding without fear to be wrong, and the freedom of the reader
> to
> insert comments and ask questions. The whole thing is very useful to all
> interested in the subjects being written about, and should not to be taken
> as
> a book in final form, which is not intended to be. Encyclopedia (18
> century)
> and Bourbaki are very important, but some fresh air is also important.
>

Thank you for the nice words (and I'm glad that -- even if nothing else
gets resolved -- at least this discussion has heightened awareness of the
existence of this project!).

The nLab (and the companion discussion forum, the nForum) are still
young and small.  It's a wiki, like Wikipedia, so that anyone can edit it.
Therefore, if you or anyone else sees flaws in an nLab article, you have
a warm open invitation to improve it!  It's easy to edit, and we appreciate
your leaving a note at the nForum to mention changes you make, or to
discuss anything you like.

We are a loosely aligned group with perhaps a dozen or so very active
contributors, including Andrew Stacey, Urs Schreiber, Zoran Skoda,
Mike Shulman, Toby Bartels, David Roberts, Tim Porter, David Corfield,
and myself. Perhaps the only things that really unite us are a belief in the
value of category theory and higher category theory, and a belief in the
value of this project.  Of course there is also Andre Joyal's CatLab,
which runs on the same easy-to-use and highly effective software.

Todd



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terminology

[Eduardo J. Dubuc]

I feel the need to clarify some of my postings.

Due to some public and private mails I realized that most people though that I
was talking about the nLab.
Well, all the time, when referring to "ghetto" or "subculture" I was aiming to
the WHOLE of the category community within mathematics, not at the nLab within
the category community.

Actually, I was not even aware of the existence of the nLab. Due to this
controversy, I visit the nLab and at first sight I essentially (not fully)
agree with Andre's comments about the nLab in his recent posting.

I say, go ahead !, nice work !

I can add that I liked the lack of solemnity and the freedom to write down
your understanding without fear to be wrong, and the freedom of the reader to
insert comments and ask questions. The whole thing is very useful to all
interested in the subjects being written about, and should not to be taken as
a book in final form, which is not intended to be. Encyclopedia (18 century)
and Bourbaki are very important, but some fresh air is also important.

I do not appreciate that a controversy about terminology be dismissed by
derision by saying

"thanks for trying to move the discussion away from terminology and back to
actual mathematical matters".

This kind of solemnity makes me shit !!

No need to move away from terminology, nobody is forbidding you to discuss
mathematics by discussing terminology, it is not one thing or the other.

We were talking about terminology, yes !!. Why not !. Terminology is
important, great mathematicians worried about it.

The "evil terminology" is wrong, somebody would even say evil, and it is
important that it does not establish itself.

This is not a fight, to abandon a terminology does not mean to loose a fight,
it just mean to become aware of some sides that were not properly considered
at the beginning. The looser is at the end the winner.

The challenge (not a minor challenge) is to find a good word "x" (or xxxx,
which means the same thing in spite to have four x's, ja!) to mean "invariant
under equivalence", or its negation, once we agree that such a word is
necessary due to the need of brevity justified by frequent use (if this
happens to be the case).

We can discuss the mathematics involved in the presence or lack of invariance
under equivalence, nobody forbids this by talking about the terminology utilized.

e.d.


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Re: terminology

[Toby Bartels]

Thorsten Altenkirch wrote in part:

>A setoid is the intensional representation of a quotient (ie
>coequalizer) and any construction involving it should respect this
>structure. To use the underlying set of a setoid to construct another
>set seems fundamentally flawed.

I would rather not say "underlying set" here, but "underlying type".
The category of types can do what it likes, but the category of sets
should already have coequalisers.  (Some type theorists do say "set" here;
I just think that it is liable to confuse category theorists.)
It is the setoids which behave like the sets that we know.

But what is flawed about using the underlying type of a set(oid)?
Group theorists who found group theory on set theory
are allowed to use the underlying set of a group.
So set theorists who found set theory (as setoid theory) on type theory
should be able to speak of the underlying type of a set(oid),
and category theorists who found category theory on type theory
should be able to speak of the underlying type of their structures.

>My understanding of an arrow category is that it's objects are the
>morphisms of  the underlying category and since this is a setoid objects
>should be represented as a setoid too.

I'm not sure what you mean by "this is a setoid".
If you mean that, given a category C (as formalised in type theory),
the morphisms of C form a setoid, then this is not true.
Given a category C and two objects x and y of C,
then the morphisms of C from x to y form a setoid, nothing more.

Even if they did form a setoid, what of that?
In Peter May's example of the category of intermediate fields
in a given field extension, the objects do form a setoid.
I call such a category a "strict category":
  http://ncatlab.org/nlab/show/strict+category
Any poset defines a strict category in which isomorphic objects are equal.
More generally, any category in which any two parallel morphisms are equal
may be made into such a strict category by defining equality as isomorphism.
Assuming an appropriate version of the axiom of choice,
any category whatsoever may be made into a strict category
by defining equality as isomorphism and making some choices
to match up hom-sets.

The fact that strict categories exist does not invalidate
the perspective from which categories are not inherently strict,
any more than the existence of monoidal categories
invalidates ordinary category theory.
Even assuming the axiom of choice,
that we can make any category into a strict category is like
our ability to make any monoidal category into a strict monoidal category;
the theory of weak categories and weak monoidal categories stands.
(Incidentally, any strict monoidal category must be a strict category,
while a weak monoidal category need not be.)

>You may say that we are only interested in objects upto isomorphism. But
>what does this mean precisely?

What it means is that, whenever anyone refers to equality of objects,
I interpret it as being a statement in strict category theory,
with all other statements being in ordinary (weak) category theory.
It is an empirical claim that the basic results of category theory
as it is normally understood do not refer to equality of objects.
It is conjecture in metamathematics that any such statement,
if a theorem, has a proof that never refers to equality of objects.
(Whether this conjecture is true or false can depend
on exactly what your foundations of mathematics are.
You also have to take care to identify defined concepts
that implicitly make reference to equality of objects.)


--Toby


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Re: terminology

[Peter LeFanu Lumsdaine]

On Tue, June 1, 2010 03:39, Thorsten Altenkirch wrote:

> A setoid is the intensional representation of a quotient (ie
> coequalizer) and any construction involving it should respect this
> structure. To use the underlying set of a setoid to construct another set
> seems fundamentally flawed.

Indeed; but what we construct is not just a set, it's a category! :-)

In Toby's construction \C |---> arr \C, the setoid structure of the
hom-sets of \C is _not_ respected if you just look at the underlying
objects of arr \C, but it _is_ respected once you look at the whole
resulting category arr \C.

This is surely no worse than the fact that in just about any construction
on setoids X |---> F(X), if you look at the underlying set of F(X), this
will not fully respect the setoid structure of X?

> My understanding of an arrow category is that it's objects are the
> morphisms of  the underlying category and since this is a setoid objects
> should be represented as a setoid too.

The trouble here is that the original setoid structure is not on the whole
arrow-set C_1, but on the individual hom-sets C_1(a,b).  The arrow
category sums this up over all a,b:C_0, and so is no longer a setoid from
this data alone.  (A dependent sum of setoids over a set doesn't have a
natural setoid structure, as far as I can see?)

In our case, of course, the object-sets _are_ also naturally setoids, with
their equalities given by isomorphisms of the categories.  But we don't
want to think of this setoid structure as primary: it's just a coarse
reflection of part of the overall category structure.

> You may say that we are only interested in objects upto isomorphism.
> But what does this mean precisely?

Going on from the above, it's an extension to the statement "we are only
interested in elements of a setoid up to the given equality relation".  So
a more precise statement could go along the lines of:  Any construction
dependent on objects should respect isomorphisms.

Best,
-p.

-- 
Peter LeFanu Lumsdaine
Carnegie Mellon University



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Re: terminology

[Thorsten Altenkirch]

Dear Toby,

Thank you for your reply. Of course I am aware of this construction.

A setoid is the intensional representation of a quotient (ie  
coequalizer) and any construction involving it should respect this  
structure. To use the underlying set of a setoid to construct another  
set seems fundamentally flawed.

My understanding of an arrow category is that it's objects are the  
morphisms of  the underlying category and since this is a setoid  
objects should be represented as a setoid too.

You may say that we are only interested in objects upto isomorphism.  
But what does this mean precisely?

Cheers,
Thorsten


On 30 May 2010, at 21:51, Toby Bartels <toby 
+categories@ugcs.caltech.edu> wrote:

> Thorsten Altenkirch wrote:
>
>> Toby Bartels wrote:
>
> [I suggested formalising category theory without equality of objects
> in intensional Martin-Löf type theory, Coq, or SEAR without equality]
>
>> I don't know any reasonable formalisation in Intensional Type Theory.
>> People usually assume that hom sets are a setoid but objects aren't.
>> This means that constructions like arrow categories are not  
>> available.
>> To avoid this one would have to formalize explicitely what is a  
>> family
>> of setoids indexed over a setoid. After this it is hard to see the
>> category theory...
>

...


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Re: terminology

[Toby Bartels]

Thorsten Altenkirch wrote:

>Toby Bartels wrote:

[I suggested formalising category theory without equality of objects
in intensional Martin-Löf type theory, Coq, or SEAR without equality]

>I don't know any reasonable formalisation in Intensional Type Theory.
>People usually assume that hom sets are a setoid but objects aren't.
>This means that constructions like arrow categories are not available.
>To avoid this one would have to formalize explicitely what is a family
>of setoids indexed over a setoid. After this it is hard to see the
>category theory...

Why do you say that one cannot construct arrow categories?
We need only dependent sums, which Martin-Löf has in his type theory,
to form the type of all morphisms of a given category C.
We cannot compare these for equality, nor do we want to.
What we do need to compare for equality are commutative squares
(which are the morphisms in the arrow category) with given corners
(actually, even with two given parallel sides), and this we can do.

To be explicit, let Ob be the type of objects of the category C;
given x, y: Ob, let x -> y be the type of morphisms from x to y.
Given further two morphisms f, g: x -> y, we have a proposition f = g.
(Martin-Löf identifies propositions with types, but Coq does not,
so I say "proposition" so you can interpret it in either system.)
Then of course, there are operations and axioms that I will skip,
except to introduce ; as notation for composition in diagrammatic order:
f: x -> y, g: y -> z |- f;g: x -> z.

Then the type of objects of the arrow category of C
is sum_{x:Ob} sum_{y:Ob} x -> y, a dependent sum of dependent sums;
a typical element of this type is (x,y,f), where x,y: Ob and f: x -> y.
Given two objects (x,y,f) and (u,v,g) of the arrow category,
the type of morphisms from (x,y,f) to (u,v,g) is
sum_{a:x->u} sig_{b:y->v} f;b = a;g. (*)
(Again, I say "sig" to keep things correct in Coq,
since then f;b = a;g is a proposition rather than a type;
this is the same as a sum to Martin-Löf.)
A typical element of this type is (a,b,p),
where p is a proof of the relevant equality.

A set theorist might well write (*) above as
{ (a,b) | a: x -> u, b: y -> v, f;b = a;g };
they do not refer to p, since set theorists accept proof irrelevance.
They would then be finished, but as type theorists,
we still need to define when parallel morphisms are equal.
We do this by imposing proof irrelevance in the definition;
that is, the definition of equality makes no reference to p.
Specifically, given parallel morphisms (a,b,p) and (c,d,q),
both from (x,y,f) to (u,v,g) in the arrow category,
we define the proposition that they are equal
to be the conjunction of a = c and b = d.
Notice that this makes sense, since a,c: x -> u and b,d: y -> v.
So we can define that equality which we need in the arrow category.

To sum up:  An object in the arrow category of C is (x,y,f),
where x and y are objects of C and f: x -> y is a morphism of C.
A morphism from (x,y,f) to (u,v,g) in the arrow category of C
is (a,b,p), where a: x -> u, b: y -> v, and p: a;g = f;b.
Finally, two such morphisms (a,b,p) and (c,d,q) are equal
if and only if a = c and b = d.  (I leave it as an exercise
for the reader to define the operations and prove the axioms
that define the arrow category of C as a category.)

If you find the summary above a bit too wordy, say
  A morphism from (x,y,f) to (u,v,g) in the arrow category of C
  is (a,b), where a: x -> u and b: y -> v such that a;g = f;b.
That we do not give a name to the proof that a;g = f;b
makes it obvious what the definition of equality of morphisms should be,
so we leave it out as an abuse of language, or a convention of definition.
If it still seems odd that it is even possible to give a proof a name,
well, that is a feature of Martin-Löf type theory and Coq
that you can ignore (just as I ignore the feature of ZFC
that it is possible to ask whether two arbitrary sets are equal),
but you can also use SEAR without equality to avoid even that.


--Toby


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Re: terminology

[Thorsten Altenkirch]

> But category theory has been done in Martin-Löof type theory:
> http://www.cs.st-andrews.ac.uk/~rd/publications/CTMLTT.pdf
> It has also been done in the type-theoretic proof assistant Coq:
> http://coq.inria.fr/distrib/v8.2/contribs-20090527/ConCaT.html
>
> In both of these, there *is* a notion of equality (or better,  
> identity)
> at all types, hence a notion of identity of objects of any given  
> category,
> allowing one to define isomorphism of categories, etc.
> However, this logicians' identity does not match mathematicians'  
> equality;
> the easiest way to see this is that there are no quotient types.
> (This means that already to do set theory, let alone category theory,
> you must define a set to be a type equipped with an equivalence  
> relation.
> Such a thing is also called "setoid", depending on which author you  
> read.)

I don't know any reasonable formalisation in Intensional Type Theory.  
People usually assume that hom sets are a setoid but objects aren't.  
This means that constructions like arrow categories are not available.  
To avoid this one would have to formalize explicitely what is a family  
of setoids indexed over a setoid. After this it is hard to see the  
category theory...

Cheers,
Thorsten

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Re: terminology

[Toby Bartels]

Colin McLarty wrote in part:

>As to articulating a way to avoid ever using identity of objects and
>identity of categories, John Baez writes

[snip]

>Has anyone yet offered a  first-order (or ML-typetheoretic)
>axiomatization of mathematics along Makkai's lines?

I don't know very much about what's been done along Makkai's lines;
I also would like to see a specific (if not final) set of axioms.

But category theory has been done in Martin-Löof type theory:
http://www.cs.st-andrews.ac.uk/~rd/publications/CTMLTT.pdf
It has also been done in the type-theoretic proof assistant Coq:
http://coq.inria.fr/distrib/v8.2/contribs-20090527/ConCaT.html

In both of these, there *is* a notion of equality (or better, identity)
at all types, hence a notion of identity of objects of any given category,
allowing one to define isomorphism of categories, etc.
However, this logicians' identity does not match mathematicians' equality;
the easiest way to see this is that there are no quotient types.
(This means that already to do set theory, let alone category theory,
you must define a set to be a type equipped with an equivalence relation.
Such a thing is also called "setoid", depending on which author you read.)

You an also use Mike Shulman's SEAR, in the variant without identity.
http://ncatlab.org/nlab/show/SEAR
http://ncatlab.org/nlab/show/SEAR#eqfree
This looks much more like the ordinary language of mathematics.
(Actually, one could modify ETCS in a similar way,
although it would no longer deserve to be called "ETCS".)


--Toby

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Re: terminology

[Colin McLarty]

Yes, Michael has said in several papers that his foundation would be
given in FOLDS.  And perhaps some specific version of the axioms has
been given in some paper that I have missed.  But I do not know of it;
and when Martin-Löf suggested last year that I might want to pursue a
type-theoretic foundation for category theory he did not mention
knowing of one existing yet.  So far as I know it remains a project.

best, Colin



2010/5/26 John Baez <baez@math.ucr.edu>:
> Colin wrote:
>
>
> As to articulating a way to avoid ever using identity of objects and
>> identity of categories, John Baez writes
>>
>>> I think Michael Makkai has done it.  He has formulated a foundational
>>> approach to mathematics based on infinity-categories, in which equality
>>> plays no fundamental role:
>>>
>>> http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf
>>>
>>> I think some approach along these general lines might ultimately become
>> quite popular.
>>
>> But so far as  know, this remains an approach, and not any specific set of
>> axioms offered as foundation.
>>
>
>
> I should let Michael speak for himself, but I have the impression that he
> intends to found all his work on FOLDS - "first-order logic with dependent
> sorts".  In this paper:
>
> http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf
>
> he writes:
>
> "The restriction on the use of equality in FOLDS is a fundamental feature.
> FOLDS is to be used in formulating categorical situations in which, for
> example, equality of objects of a category is not an admissible primitive.
> The absence of term-forming operators, to be interpreted as
> functions, is a consequence of the absence of equality; it seems to me that
> the notion of "function" is incoherent without equality.
>
> It is convenient to regard FOLDS a logic without equality entirely, and deal
> with equality, as much as is needed of it, as extralogical primitives."
>
> Best,
> jb
>


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Re: terminology

[John Baez]

Colin wrote:


As to articulating a way to avoid ever using identity of objects and
> identity of categories, John Baez writes
>
>> I think Michael Makkai has done it.  He has formulated a foundational
>> approach to mathematics based on infinity-categories, in which equality
>> plays no fundamental role:
>>
>> http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf
>>
>> I think some approach along these general lines might ultimately become
> quite popular.
>
> But so far as  know, this remains an approach, and not any specific set of
> axioms offered as foundation.
>


I should let Michael speak for himself, but I have the impression that he
intends to found all his work on FOLDS - "first-order logic with dependent
sorts".  In this paper:

http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf

he writes:

"The restriction on the use of equality in FOLDS is a fundamental feature.
FOLDS is to be used in formulating categorical situations in which, for
example, equality of objects of a category is not an admissible primitive.
The absence of term-forming operators, to be interpreted as
functions, is a consequence of the absence of equality; it seems to me that
the notion of "function" is incoherent without equality.

It is convenient to regard FOLDS a logic without equality entirely, and deal
with equality, as much as is needed of it, as extralogical primitives."

Best,
jb


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Re: terminology

[Reinhard Boerger]

Dear Colin, dear all,

you wrote:

> It is an interesting impulse in higher category theory to avoid
> identity in favor of isomorphism on the level of objects, and to avoid
> isomorphism in favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.

As far as I remember I listened to a talk by a logician called H. Preller in
the seventies. She developed a language for categories, which did not
contain the identity of objects.


Greetings
Reinhard



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Re: terminology

[Toby Bartels]

Vaughan Pratt wrote in part:

>Moreover most of us would agree that the proposition "the prime factors
>of M = 7^7^7^7 + 5^5^5^5 + 1 (7#4 + 5#4 + 1 where m#n denotes an
>exponential stack of n m's) are all greater than 2 billion and there are
>more than a thousand distinct such" not only makes perfect sense but is
>either true or false.  However fewer might be willing to join me in
>insisting that it is certainly true.

Since I know very little about these issues,
I'm not ready to accept your claim that it is true.
(I know that you sketched a way for me to verify it
by performing some calculations on my laptop,
but it would take a while for me to figure out what to program
and then to convince myself that the output meant what you say.)

However, I am happy to agree that the statement is true or false.

>Those who question excluded middle for this proposition may have
>received different wisdom about N than the rest of us, though if I'm
>right then there's a constructive proof of the proposition that can be
>checked on any laptop in under an hour, which should then oblige the
>intuitionistic objectors to stand down.

Anyone who doubts excluded middle for *this* proposition
is not merely a constructivist, or even an intuitionist.
Excluded middle for this proposition is provable in Heyting arithmetic.
While a straightforward calculation of the factors of M
would not fit into the physical universe, it is still finite.

Those who doubt excluded middle (or meaningfulness) for this proposition
go beyond intuitionism; they have been called "ultra-intuitionists",
although the preferred term these days is "ultra-finitists".
As someone who is quite comfortable with constructivism,
I still find ultra-finitism a very strange way to think.

Ultra-finitists definitely have a different recieved wisdom about N
from what the rest of us have received.

Ob categories:  Does anybody know any work on ultra-finitism
from the perspective of categorial logic? (somewhat in the way
that topos theory can provide a perspective on constructivism).
I doubt that any exists, but I would it would be nice if it did.


--Toby


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Re: terminology

[John Baez]

Colin wrote:

It is an interesting impulse in higher category theory to avoid identity in
> favor of isomorphism on the level of objects, and to avoid isomorphism in
> favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.
>

I think Michael Makkai has done it.  He has formulated a foundational
approach to mathematics based on infinity-categories, in which equality
plays no fundamental role:

http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf

I think some approach along these general lines might ultimately become
quite popular.  However, to think in an easy intuitive way about a
mathematical world without equality, we need new definitions of words such
as "the" and "is". Those who find it unpleasant to change the definition of
words such as "autonomous" may think it absurd to consider a such a radical
shift in basic terminology. However, we can already see these words changing
their meanings as we pass from reasoning within sets - where we say "the"
product 2 x 3 "is" 6 - to reasoning within categories - where we say "the"
product of "the" 2-element set and "the" 3-element set "is" "the" 6-element
set.

For a readable introduction to some of Makkai's ideas, try:

http://www.math.mcgill.ca/makkai/equivalence/equivinpdf/equivalence.pdf

Best,.
jb

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Re: terminology

[Vaughan Pratt]

On 5/23/2010 8:39 AM, Colin McLarty wrote:
> It is an interesting impulse in higher category theory to avoid
> identity in favor of isomorphism on the level of objects, and to avoid
> isomorphism in favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.

Is identity even definable?  I thought it was a kind of received wisdom,
like the natural numbers.  All of us seem to be working with the same
notions of = and N, but what are they, exactly?

That was only intended as a rhetorical question, btw.  We can readily
agree on some properties of = and N, which logicians of various stripes
have gone to the trouble of spelling out, and which number theorists
both analytic and algebraic have expanded on.

Moreover most of us would agree that the proposition "the prime factors
of M = 7^7^7^7 + 5^5^5^5 + 1 (7#4 + 5#4 + 1 where m#n denotes an
exponential stack of n m's) are all greater than 2 billion and there are
more than a thousand distinct such" not only makes perfect sense but is
either true or false.  However fewer might be willing to join me in
insisting that it is certainly true.

Those who question excluded middle for this proposition may have
received different wisdom about N than the rest of us, though if I'm
right then there's a constructive proof of the proposition that can be
checked on any laptop in under an hour, which should then oblige the
intuitionistic objectors to stand down.

(No, I don't currently know a single prime factor of M and I don't
believe anyone else does either.  I do however know the least prime
factors of both M + 1 and M + 958; leaving the former as an exercise,
the latter is 1,985,781,901.  M in decimal is 1755522...1375469 where
the number of omitted digits when itself written in decimal has 695,975
digits, so although M in binary wouldn't fit in the universe let alone a
laptop's random-access memory its length in binary would easily fit in
the latter.  The requisite calculations for all these observations take
only minutes on an ordinary laptop.)

Without exponentiation in the language, M would not be known to us: with
only the polynomial operations the requisite expression 7*7*...*7 +
5*5*...*5 + 1 would stretch beyond the farthest known galaxies.  This
question about M, which is a question about N, could therefore not have
arisen.  With it, the question becomes part of our understanding, or
lack thereof, of N.

The same can be said of identity.  The richer the language, the more
tools we have to probe our understanding of identity, and the clearer
our lack of complete understanding of it becomes.

Vaughan Pratt

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Re: terminology

[Colin McLarty]

I have very much appreciated André's subtlety on this issue in conversation

2010/5/22 Joyal, André <joyal.andre@uqam.ca>:
> I wrote
> ------------------------------------------------------------
> One lies in the fact that equivalent categories are considered
> to be the "same",
> -------------------------------------------------------------
> I was careful not to write
> ------------------------------------------------------------
> One lies in the fact that equivalent categories are considered
> to be the same,
> -------------------------------------------------------------

John Baez has written carefully on this point too.

But not everyone is so careful and Ronnie has good reason to be
concerned about a tendency to sweep away distinctions that do need to
be made.

Isomorphic categories too must be distinguished from one another, some
times and for some purposes notably including all currently
articulated versions of categorical foundations.

Grothendieck gave it a fine nuance in Tohoku (p. 125) saying "Aucune
des equivalences de categories qu'on rencontre en pratique n'est un
isomorphisme (none of the equivalences one meets in practice are
isomorphisms)."  He stressed that we must distinguish isomorphisms
from equivalences.  Throughout that and later works he *constructs* a
great many categories up to isomorphism, and not just up to
equivalence.  We do not meet these isomorphisms, we construct them --
and it is quite important that once constructed they are not merely
equivalences.

It is an interesting impulse in higher category theory to avoid
identity in favor of isomorphism on the level of objects, and to avoid
isomorphism in favor of equivalence on the level of categories.   But
so far as I know no one has yet articulated a way to avoid ever using
identity of objects and identity of categories.

Colin






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Re: terminology

[Joyal, André]

Dear Ronnie,

I totally agree with what you wrote.

I wrote 

------------------------------------------------------------
One lies in the fact that equivalent categories are considered 
to be the "same",
------------------------------------------------------------- 


I was careful not to write

------------------------------------------------------------
One lies in the fact that equivalent categories are considered 
to be the same,
------------------------------------------------------------- 

Sorry for not been clear enough.
I hope this settle our apparent disagreement.

Best,
André




-------- Message d'origine--------
De: Ronnie Brown [mailto:ronnie.profbrown@btinternet.com]
Date: sam. 22/05/2010 17:43
À: Joyal, André
Cc: Urs Schreiber; categories@mta.ca
Objet : Re: RE : categories: Re terminology:
 
Dear André

There seems to me to be a tremendous amount of great work going on 
higher category theory, but when you write

-----------------------------------------------------
One lies in the fact that equivalent categories are considered to be the 
"same", 

even if [or] when they are not isomorphic.
-----------------------------------------

this seems to go against the grain of what I have been doing in groupoids since I decided they were valuable in about 1965! It sounds like the old canard `groupoids reduce to groups', so there must be some confusion in my mind on what you are saying. 


....



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Re: terminology

[Ronnie Brown]

Dear André

There seems to me to be a tremendous amount of great work going on 
higher category theory, but when you write

-----------------------------------------------------
One lies in the fact that equivalent categories are considered to be the 
"same", 

even if [or] when they are not isomorphic.
-----------------------------------------

this seems to go against the grain of what I have been doing in groupoids  since I decided they were valuable in about 1965! It sounds like the old  canard `groupoids reduce to groups', so there must be some confusion in my mind on what you are saying. 

One thing that took me a while to realise was that it was not enough to study the fundamental groupoid or a fundamental group but one needed to consider intermediate cases, namely the fundamental groupoid on a set of base points chosen according to the geometry at hand. (`Algebraic topology'  has not understood this it seems.) The vertices of a groupoid give a spatial component to group theory, a kind of geography, and sometimes, even often, that is needed to model the geometry.  So for example it is useful  to replace the trefoil group which has 2 generators x,y and one relation  x^2=y^3 by the trefoil groupoid which is the double mapping cylinder (homotopy pushout) in groupoids of the two maps Z \to Z, given by squaring  and cubing. So we add an extra groupoid generator iota on different vertices which turns x^2 into y^3. This corresponds to the double mapping construction to give a CW-complex. 

So groupoids give the strict algebra of keeping the information which makes things the same. 

In higher dimensions we want not just commutative diagrams but control of  the ways of filling these diagrams. If the diagram is a pentagon (as we all know does happen) I would want a pentagon as part of the geometry, and the only question is how to deal with multiple compositions of various such objects, and that was the aim of David Jones thesis on Polyhedral T-complexes. The point is that the pieces to be composable have to be all faces but one of a general poyhedral `horn', the process of composing them  is the filler of the horn, and the composite of the pieces  is the remaining face of the filler. (It was not attempted to do this in category rather than groupoid terms, and that is still a mystery!) So you can see I have long been very sympathetic to using the Kan condition for describing algebraic or structural objects, but find the simplicial approach too awkward (for me, of course; I found the way Nick Ashley coped with that was amazing). 

I do not want to consider equivalent groupoids the same, as I may want to  use the spatial components to describe how they might be glued together.  It is partly the old tag of not throwing away information till the last possible moment. 

On the other hand, some computations are best done at the strict level, rather than the weak one. I mention here the rotations in my paper:

``Higher dimensional group theory'', in {\em Low dimensional topology},  London Math Soc. Lecture Note Series 48 (ed. R. Brown and T.L.  Thickstun, Cambridge University Press, 1982), pp. 215-238.

(see also a fuller exposition in the new book on Nonabelian algebraic topology), which would seem to be more difficult to write out at the lax level. The fact that the strict calculations imply the existence of certain homotopies is part of the interest. 

So in the work with Higgins a Kan fibration - from the singular filtered complex of a filtered space to the quotient to give a strict structure - ties in the lax and the strict in a necessary way for the theory and calculations. 

I am really searching for points of agreement. 

Best regards

Ronnie


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Re: terminology

[Eduardo J. Dubuc]

soloviev@irit.fr wrote:
> My personal opinion is that this process is very much influenced
> by the pressure of "bibliometry", "impact factors" and other "modern
> trends" - people often not very scrupulously invent and reinvent
> terminology to be better cited, and, conscious or not, it often very
> much smells of imposture.
>
> Sergei Soloviev


I agree with this. But it should be clear that many times it is not 
conscious, but certainly it often smells of imposture. Other times it 
smells of excessive logic and formalism.

Concerning "injective" I see no problem at all that some times injective
means (1-1) and other times it means the dual of projective.

Where is the problem !!, the context always tells you which meaning it is
being used.

e.d.


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Re: terminology

[Peter Selinger]

I had written:
> 
> My last comment is that, unlike what Jeff Egger claimed, "autonomous
> category" is not a special case of "*-autonomous category", because no
> symmetry is assumed in autonomous categories. Unless of course one
> first drops symmetry from the definition of *-autonomous categories,
> as Jeff has also suggested. As it stands, neither of "autonomous" and
> "*-autonomous" implies the other, which is perfectly fine in my
> opinion, since they are two different words.

I would like to clarify that Jeff himself did not say anything false,
because in the context in which he said it, he had in fact assumed the
non-symmetric definition of *-autonomous category (of [Barr 1995]).
Sorry if it sounded like I was accusing him.

My intention was only to point out that the statement "autonomous
categories are a special case of *-autonomous categories" cannot be
quoted out of context, because it is false under the original
definition of *-autonomous category that includes symmetry (of [Barr
1979]). Since it had already been quoted out of context when I wrote
the above, I just wanted to point out how the potential confusion. 

I think this is a very apt illustration of what happens if a term with
an existing meaning is redefined to mean something else. Henceforth it
is impossible for anybody to use the term (with either meaning)
without first giving a definition. That's no problem in a math paper,
where definitions are usually given or cited anyway, and therefore
terminology is in principle arbitrary. But it does tend to hobble
everyday discussion.

-- Peter

P.S.: since I have a demonstrated ability to put my foot in my mouth,
I'd like to clarify that I am not accusing Mike Barr of anything
either. His 1995 paper is clearly entitled "Non-symmetric *-autonomous
categories", and the inside of the paper clearly explains the
distinction. It is only in subsequent use that any confusion arises.
The usual solution, of putting either (non-symmetric) or (symmetric)
in parentheses the first time the term is used, and omitting it for
subsequent uses, is perfectly adequate. I am very happy with the
statement "an autonomous category is a special case of a
(non-symmetric) *-autonomous category".

M. Barr (1979). "*-Autonomous Categories", Lectures Notes in
Mathematics 752. Springer. 

M. Barr (1995). "Non-symmetric *-autonomous categories". 
Theoretical Computer Science 139:115–130.


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Re: terminology

[wlawvere]

Dear Eduardo and everybody:

In one of your papers you used the term
Nullstellensatz for a special case (in some
sense an "algebraically closed"case). I
propose to use that term in this more
general case.

The parameters for various
traditional cases can be perhaps expessed
by an essential connected morphism of
toposes E->S. That is, a full inclusion of
"relatively discrete" into "relatively
 continuous" which has both left adjoint
("connected components") and right adjoint
("points").

In that context there is a natural map from
points to components; if it is epic, we can
say that the Nullstellensatz holds for
E->S.

If S is just the category of abstract sets,
one could think of E as algebraically closed if
the Nullstellensatz holds.

But as seems implicit in Galois theory, for
algebraic geometry over a non-algebraically
closed K, the appropriate base topos S consists
not of abstract sets, but rather of sheaves
on C = the opposite of the category of finite
extensions of K, with every map covering. If E
is the topos of sheaves on (finitely generated
K-algebras )^op with respect to a topology that
restricts to the above on C, I believe
we have a classical example of both your
formulation and mine.

Bill

PS There are other stronger results that also
could be called  Nullstellensatz, involving
another topos F between E and S, such as
the one generated by algebras that are finite
dimensional as K-vector spaces, or one
suggested by Birkhoff's SDI theorem. What
is the appropriate statement for these results ?


Quoting Eduardo Dubuc <edubuc@dm.uba.ar>:

> hello:
>
> Given a set CC of objects in a topos EE, consider the following
> property:
>
>       " X no= empty  iff  exists C \in CC, hom(C, X) no= empty "
>
> example; CC = a set of generators
>
> Has (this property) already  a name ?
>
> If not, can you suggest one ?
>
> Any answer will be welcome.
>
> (Notice that if CC is a set of points (instead of objects) we say
> that
> there are enough points)
>
> Thanks          Eduardo J. Dubuc
>
>
>
>

terminology

[Eduardo Dubuc]

hello:

Given a set CC of objects in a topos EE, consider the following property:

      " X no= empty  iff  exists C \in CC, hom(C, X) no= empty "

example; CC = a set of generators

Has (this property) already  a name ?

If not, can you suggest one ?

Any answer will be welcome.

(Notice that if CC is a set of points (instead of objects) we say that
there are enough points)

Thanks          Eduardo J. Dubuc

Re: terminology

[Eduardo Dubuc]

I have been asked why I reacted to the intended reeplacement of the names
"cartesian and cocartesian"  by  "prone and supine".  I have given several
reasons, but the one underlying the whole issue is the following:

The reason is that since a long time I have been worried about the ghetto
(in the sense of being isolated from the rest) characteristic of a certain
category theory community (or group of people). And P. May has reacted
concerning "prone and supine" probably because of reasons related to this.

The mathematical community  have been using "cartesian and cocartesian"
since always, and the introduction of "prone and supine" inside this
group will confirm even more the isolation. Examples abound, see M.Barr
introduction of "Molecular topos" to replace Grothendieck's "Locally
connected topos".

No matter how many linguistic points in favor a given name may have (like
prone and supine), to replace a well stablished  name intoduced by a
great mathematician (or school of mathematics) only puts you in
ridiculous.

P. May probably was feeling somehow that this will be extended by the
mathematical community to all category theory practicioners.

I profit by this mail to mention that concerning the concept "final" and
"initial", I am happy (and not surprised) to learn that these words have
been used since a long time to indicate the same categorical concept that
myself, and will certainly refer to the indicated bibliography to further
justify my use of these words.

Re: terminology

[vs27]

On Dec 29 2005, Vaughan Pratt wrote:

> Without taking sides on the prone/supine terminology question, I do have
> a strong reaction to the Benabou/May/Dubuc concern that respect for a
> field is undermined by its adoption of frivolous terminology.
>
Dear Vaughan, as everybody has a say. Just my views.
I prefer some nomenclature that sounds mathematical,
rather than based on the name of a friend or a private joke.
(may be i don't understand all the jokes ?)
Also in any case one should avoid  renaming existing
concepts, that is just not fair.


Good opportunity to wish happy new year to everybody.

Re: terminology

[Eduardo Dubuc]

We should not put everything in the same bag !!

"strange," "charm," "beauty" and even "quark" itself

are beautiful  and poetic names  to refer to objects or concepts which
precisely we do not want to associate any precise meaning in everyday
language, and on the other hand, the objects or concepts are  introduced
whith those names.

"prone/supine"  are all the contrary, they intent to reflect in everyday
language just one aspect of an existing concept which has many, and more
important, they are used in place of a well stablished name.

all this has nothing to do with  "young field" as opposed to "mature
subject"

silly names (if any) in physics would be as bad as in any other subject

do not confuse  things, I found  the  "Scott is sober" an exelent example
of humor that does not undermine respect for the field. Another exelent
example that comes to my mind is M. Barr's "The point of the empty set"

edubuc

>
> Without taking sides on the prone/supine terminology question, I do have
> a strong reaction to the Benabou/May/Dubuc concern that respect for a
> field is undermined by its adoption of frivolous terminology.
>
> This may be a valid concern for a young field like category theory, but
> for a more mature subject such as physics, a more relevant concern is
> the undermining of the ability to poke fun at oneself by the fear of not
> being taken seriously.
>
> Has the adoption of frivolous nomenclature for quarks ("strange,"
> "charm," "beauty" and even "quark" itself) diminished in any way the
> world's respect for quarks and their investigators?
>
> And what of computational topology?  Should we turn a blind eye to
> whether Scott is sober, and substitute a more genteel euphemism for his
> bottom?
>
> Vaughan Pratt
>
>

Re: terminology

[Nikita Danilov]

Vaughan Pratt writes:
 > Without taking sides on the prone/supine terminology question, I do have
 > a strong reaction to the Benabou/May/Dubuc concern that respect for a
 > field is undermined by its adoption of frivolous terminology.
 >
 > This may be a valid concern for a young field like category theory, but
 > for a more mature subject such as physics, a more relevant concern is
 > the undermining of the ability to poke fun at oneself by the fear of not
 > being taken seriously.
 >
 > Has the adoption of frivolous nomenclature for quarks ("strange,"
 > "charm," "beauty" and even "quark" itself) diminished in any way the
 > world's respect for quarks and their investigators?

There indeed are drawbacks whenever scientific terms are contrary to the
centuries old tradition not taken from Greek or Latin languages (that,
thanks to their very regular and flexible system of word formation are
so suitable for taxonomies) shared by many cultures. For one thing,
words of existing languages are not in one to one mapping, and then a
term from contemporary language may be not culturally neutral (consider
silly naming wars for transuranium elements).

On the other hand, I stopped using "co-product" after more than one
person with the background in classical languages read it as
"copro-duct".

 >
 > And what of computational topology?  Should we turn a blind eye to
 > whether Scott is sober, and substitute a more genteel euphemism for his
 > bottom?
 >
 > Vaughan Pratt

Nikita.

Re: terminology

[Vaughan Pratt]

Without taking sides on the prone/supine terminology question, I do have
a strong reaction to the Benabou/May/Dubuc concern that respect for a
field is undermined by its adoption of frivolous terminology.

This may be a valid concern for a young field like category theory, but
for a more mature subject such as physics, a more relevant concern is
the undermining of the ability to poke fun at oneself by the fear of not
being taken seriously.

Has the adoption of frivolous nomenclature for quarks ("strange,"
"charm," "beauty" and even "quark" itself) diminished in any way the
world's respect for quarks and their investigators?

And what of computational topology?  Should we turn a blind eye to
whether Scott is sober, and substitute a more genteel euphemism for his
bottom?

Vaughan Pratt

Re: Terminology

[Eduardo Dubuc]

I very strongly agree with J. Benabou's comments about "prone" and
"supine", and P. May's opinion that "I'd like category theory no longer to
be regarded as nonsense in this country --- it still is in many quarters,
as I could easily prove --- and such terminology is not exactly helpful to
the cause!"

I recall P. Johnstone that he himself named his book  "Elephant Book"
because every body has different version of what a topos is, reflecting
only one of the many aspects of the concept.

Names like "Prone" and "Supine" correspond (with luck) to only one of the
many aspects of the concept of cartesian and its dual (in a sense)
cocartesian.

Also, there is a clear ethical aspect involved when a stablished
terminology that has been historically introduced by particular people
suffers a move to be eliminated and reeplaced by another.

But, coming back to the question above, i am also against the habit to
name a new mathematical concept with words that have a precise meaning in
everyday language (as prone, supine, etc).

Presisely, I do not know what does it mean exactly "Cartesian" (has
something to do with Descartes ...), but I know presisely what it is a
"Cartesian arrow" (in mathematics).

Colorful terminology taken from everyday language is an strong indication
to serious mathematicians that the subject should no be taken seriously
(see for example the claims of  "Catastrofe Theory" as opposed to the
sober "Classification of singularities of C-\infty mappings", and a lot of
similar examples).

As P May points out, "  . . .  such terminology is not exactly helpful to
the cause!".

The meaning of a mathematical concept should be given by the concept
itself, and not by the connotation that its name has in everyday language.

Terminology

[jean benabou]

I have seen in this mail that the suggestion of Taylor and Johnstone to
replace cartesian and cocartesian maps by prone and supine ones begins to
be accepted. When I first saw that suggestion, I was so amazed that I
thought it was a joke, and not such a good one. I still hope it is no more
than that. But, just in case, and before it is too late, I want to say
that I am very strongly opposed to such changes for many reasons:
linguistic, mathematical, and ethical, which I am ready to explain in
detail if I am asked to do so.

Re: terminology

[Marco Grandis]

In reply to Stasheff's question on terminology for homotopy coherent algebras:

>but now what about e.g. 1-homotopy associaitve satisfying a STRICT
>pentagon??
>perhaps strict 1-homotopy

I would say:

"2-strict sha-algebra", as motivated below.
(sha = strongly homotopy associative)

However: after the strict pentagon, this structure has a second coherence
condition for the associativity homotopy
(which disappears for monoidal categories, just because their 2-morphisms
are trivial)

____________

In a paper [*] on strongly homotopy associative (differential) algebras, I
proposed this definition (4.2; pages 38-39).

Notation: a sha-algebra is a graded module  A  with morphisms
(sort of components of a global differential  d of bar coalgebras)

   d_1: A --> A  (degree - 1; the differential)

   d_2: AoA --> A  (degree 0; the product)

   d_3: AoAoA --> A  (degree 1; the associativity 1-homotopy)
   ........
   d_n: A^n  -->  A   (degree  n - 2; the coherence n-homotopy)
   ........

( o = tensor product;  ^n  = tensor power)

under axioms

(1)  d_1.d_1 = 0
(2)  ....

(expressing  dd = 0  for the global differential).

DEF. This is called an *n-strict sha-algebra* if  d_p = 0  for  p > n.

Equivalently,  the morphisms  d_1,..., d_n  have to satisfy the original
axioms (1)  ... (n)
plus n - 1 conditions obtained from the axioms  (n+1) ... (2n - 1),
cancelling the null  d_p's
(the remaining axioms become trivial).

This gives:

1-strict = differential module

2-strict = associative differential algebra

3-strict = 1-homotopy associative differential algebra
   with strict pentagon (from axiom (3)) and axiom (4) reduced to:

   (4)   d3 (1o1od3 + 1od3o1 + d3o1o1) = 0.

_______

So far in that paper.
The name is chosen to make  d_n  the last relevant component, in the
n-strict case.

I might now (more geometrically) prefer a - 1 shift in these names, so that
the last example would be named 2-strict, in accord with the fact that the
last relevant homotopy is a an ordinary ("one-dimensional") homotopy and
everything becomes strict starting with "dimension 2".

_______

Reference:

[*] M. Grandis, On the homotopy structure of strongly homotopy associative
algebras, J. Pure Appl. Algebra 134 (1999), 15-81.

_______

Regards    MG

terminology

[James Stasheff]

In `higher homotopy theory', terminology has not setled down nor is it
transparent

homotopy ___________ algebra can mean a variety of things

letting ______________ = associative
it can mean JUST that there is a homtopy for associaitivity
or
some authors use it to mean A_\infty

which I initially tried to indicate by strongly homtopy associative

_\infty seems to have caught on to mean the presence of higher homtopies
of all orders

in most but not all cases, such algebras have a homtopy invariant
defintion

so I would suggest the following revisionist terminology

1-homotopy associative means JUST that there is a homotopy for
associaitivity

similarly n-homotopy associative would mean homotopies of homotopies
of...

homotopy invariant ___ algebra would mean just what it says

so far so good
but now what about e.g. 1-homotopy associaitve satisfying a STRICT
pentagon??

perhaps strict 1-homotopy

open to suggestions



	Jim Stasheff		jds@math.upenn.edu

		Home page: www.math.unc.edu/Faculty/jds

As of July 1, 2002, I am Professor Emeritus at UNC and
I will be visiting U Penn but for hard copy
        the relevant address is:
        146 Woodland Dr
        Lansdale PA 19446       (215)822-6707

Terminology

[Krzysztof Worytkiewicz]

Dear Categories,

Is it  established terminology  to call *injective* a faithful functor
which is injective (in the usual sense) on objects ?

Cheers, Krzysztof

Re: Terminology

[Max Kelly]

In response to Jean Benabou's question about the terminology for what some
call "cofinal" functors, may I refer him to Section 4.5 of my book "Basic
Concepts of Enriched Category Theory", where such notions are considered
in considerable generality? In so far as we deal with functors - meaning
"V-functors" in the context of V-enriched category theory - the terms I
used, which are those common here at Sydney, are "final functor" and
"initial functor". These notions, however, make sense only when V is
cartesian closed; for a more general symmetric monoidal closed V, what is
said to be initial is a pair (K,x) where K is a V-functor A --> C and x is
a V-natural transformation H --> FK, where H: A --> V and F: C --> V are
V-functors with codomain V, and thus are "weights" for weighted limits.
The 2-cell x expresses F as the left Kan extension of H along K if and
only if, for every V-functor T: C --> B of domain C, the canonical
comparison functor (induced by K and x) between the weighted limits, of
the form

                   (K,x)* : {F,T} ----> {H,TK},
		   
is invertible (either side existing if the other does); the book contains
a third equivalent form making sense whether the limits exist or not. When
these equivalent properties hold, the pair (K,x) is said to be INITIAL.
The point is that, in this case, the F-weighted limit of any T can be
calculated as the H-weighted limit of TK.

When V is cartesian closed, we have for each V-category C the V-functor C
---> V constant at the object 1, limits weighted by which are the CONICAL
limits, which when V = Set are the classical limits. For such a V we can
consider the special case of the situation considered above, where each of
H and F is the functor constant at the object 1, and where x is the unique
2-cell between H and FK; we call the functor K "initial" when this pair
(K,x) is so; equivalently when the canonical lim T ---> lim TK is
invertible for every T (for which one side exists -- or better put in
terms of cones), or equivalently again when

           colim C(K-,c) == 1  for each object c of C.
	  
When V = Set, this is just to say that each comma-category K/c is
connected. When the category C is filtered, a fully-faithful K: A --> C is
final (dual to initial) precisely when each c/K is non-empty.

The book goes on to discuss the Street-Walters factorization of any (ordinary)
functor into an initial one followedby a discrete op-fibration.	

The above being so, it seems that Jean's good taste has led him to suggest
the very same nomenclature that recommended itself to us at Sydney.  I
should have been happier, though, if he had recalled the treatment I gave
lovingly those many years ago. There are many other expositions in the
book that I am equally happy with, and which I am sure Jean would enjoy.
By the way, someone spoke recently on this bulletin board of the book's
being out of print and hard to get; I've been meaning to find the time to
reply to that, and discuss what might be done. The copyright has reverted
to me; but the text does not exist in electronic form - it was written
before TEX existed, and prepared on an IBM typewriter by an excellent
secretary with nine balls.

I suppose I could have some copies - one or more hundreds - printed from
the old master, after correcting the observed typos. But the photocopying
and binding and the postage would cost a bit. I'ld be happy to receive
suggestions, especially from such colleagues as would like to get hold of
a copy. By the way, I sent out preprint copies to about 100 colleagues
back in 1980 or 1981; if any of those are still around, I point out that
they contain the full text. So too do those copies which appeared in the
Hagen Seminarberichte series. Once again, I look forward to any comments,
either in favour of or against making further copies.

Max Kelly.

Re: Terminology

[Dr. P.T. Johnstone]

> I am confronted with problems of "contradictory terminology" which I would
> like to solve and, since english is not my language, I need some
> suggestions.
> Let  F: Y-----> X be a functor such that for every object  x of  X the comma
> category  (x,F) is connected.Such functors, although they are not defined in
> all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
> handbook (Vol.1-'2.11-p.69) but none of these terms is satisfactory.
>  The "cofinal" name comes obviously from the vocabulary of ordered sets
> which are special cases, but in category theory  "co" is now associated with
> dual notions.

There was some discussion of this point on the categories mailing list
a year or two back. I think there was general consensus that the "co"
in "cofinal" was redundant, and that such functors should simply be
called "final". This is the term used in Mac Lane's book (section IX 3,
p.217) -- I believe Mac Lane was the first to shorten "cofinal" to "final".
For some reason, Borceux chose to use the opposite convention regarding
"initial" and "final" in his book (although, in Exercise 2.17.8 on page 94,
he seems to have reverted to the same convention as Mac Lane).

Peter Johnstone

Re: Terminology

[Steve Lack]

Jean Benabou writes:
 > I am confronted with problems of "contradictory terminology" which I would
 > like to solve and, since english is not my language, I need some
 > suggestions.
 > Let  F: Y-----> X be a functor such that for every object  x of  X the comma
 > category  (x,F) is connected.Such functors, although they are not defined in
 > all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
 > handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory.

Mac Lane calls such functors ``final'' in Categories for the Working
Mathematician. I do too.

Steve Lack.

Terminology

[Jean Benabou]

I am confronted with problems of "contradictory terminology" which I would
like to solve and, since english is not my language, I need some
suggestions.
Let  F: Y-----> X be a functor such that for every object  x of  X the comma
category  (x,F) is connected.Such functors, although they are not defined in
all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory.
 The "cofinal" name comes obviously from the vocabulary of ordered sets
which are special cases, but in category theory  "co" is now associated with
dual notions.
The "initial" name is even less satisfactory, because:
(i) If  Y=1, F is identified with an object  x of X and F is "initial" iff x
is a terminal object of X  !
(ii) More generally, if Y has a terminal object  t  then F is "initial" iff 
F(t) is terminal !
(iii) Even more generally yet, without assuming the existence of terminal
objects in Y or X :
 Let X^ and Y^ be the categories of presheaves on X and Y, and  F! :X^----->
Y^  the canonical extension of F to these categories.If  T is the terminal
object of Y^ one can easily show that  F has the previous property iff 
F!(T) is terminal in X^.(Which by the way, gives the nicest proof of the
stability under composition of such functors)
I propose to call these functors either "terminal" or better "final" but I
would like to know if this would not conflict with previous terminology.
Thanks for your help. 

re: terminology

[James Stasheff]

So far the clear front runner is `face complex'
Thanks to all the nominators.

Grandis points out why the historical semi-simplicial
won't fly for at least another generation.

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds

Re: terminology

[Marco Grandis]

J. Stasheff wrote:

>Has terminology settled down?
>I can recall seeing various terms for
>``simplicial object without degeneracies''


I am afraid it has not.

In my opinion, it should be called 'semi-simplicial object', consistently
with the original terminology in Eilenberg-Zilber (see references below).
Such a term has been adopted in Weibel's text on homological algebra
(1994). But there seems to be some opposition.
___

I hope the following reconstruction of terminology is correct.

1. What is now called a simplicial object was introduced by Eilenberg and
Zilber (1950); they use:

(a) [already existing] 'simplicial complex' = set with distinguished parts;
(b) [new term] 'semi-simplicial complex' = graded set with faces;
(c) [new term] 'complete s.s. complex' = graded set with faces and degeneracies;

2. Later, notion (c) was recognised as more important than (b) and called
'semi-simplicial complex', leaving (b) without any standard name.

3. Since May's book (1967) at least, notion (c) gradually settled down as
'simplicial set', generalised to 'simplicial object' in a category; this is
now standard.

4. It should now be natural to use a similar term, 'semi-simplicial object
(possibly: set)' for (b), i.e. a 'simplicial object without degeneracies'
(as in Weibel 1994). This is consistent with the original use in
Eilenberg-Zilber and gives a non-ambiguous set of terms for the three
notions recalled:

(a) 'simplicial complex' (also: combinatorial complex)
(b) 'semi-simplicial object (set)'
(c) 'simplicial object (set)'

However, I used myself this terminology in a paper published in '97 and had
strong reactions from people attached to the terminology in use between
50's and '60s (point 2 above).

___

References:

S. Eilenberg - J.A. Zilber, Semi-simplicial complexes and singular
homology, Ann. of Math. 51 (1950), 499-513.

J.P. May, Simplicial objects in algebraic topology, Van Nostrand 1967.

C.A. Weibel, An introduction to homological algebra, Cambridge Univ. Press,
Cambridge, 1994.

___

With best regards

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752

http://www.dima.unige.it/STAFF/GRANDIS/
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/

Re: terminology

[Paul Glenn]

How about "face complex"?

James Stasheff wrote:
> 
> Has terminology settled down?
> I can recall seeing various terms for
> ``simplicial object without degeneracies''
> 
> .oooO   Jim Stasheff            jds@math.unc.edu
> (UNC)   Math-UNC                (919)-962-9607
>  \ (    Chapel Hill NC          FAX:(919)-962-2568
>   \*)   27599-3250
> 
>         http://www.math.unc.edu/Faculty/jds

-- 
Paul Glenn
Department of Mathematics
Catholic University of America

terminology

[James Stasheff]

Has terminology settled down?
I can recall seeing various terms for 
``simplicial object without degeneracies''

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds