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: 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: 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: 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?=

[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: 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: 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]

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: 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: 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: 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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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: 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?=

[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?=

[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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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: 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: 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 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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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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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?

[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

[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: 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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bilax monoidal functors

[David Yetter]

John Baez could not recall whether bilax and Frobenius monoidal functors =
are the same.

The answer is no, in the usage I'd been familiar with,  bilax meant =
simply equipped with both lax and oplax structures, while a Frobenius =
monoidal functor satisfies  additional coherence relation which =
generalize the relations between the multiplication and comultiplication =
in a Frobenius algebra.

A bilax monoidal functor from the one-object monoidal category to VECT =
would be a vector-space with both an algebra and a coalgebra structure =
on it (no coherence relations relating them), while a Frobenius monoidal =
functor would be a Frobenius algebra. =20

Aguiar (with good reason), on the other hand, reserves bilax for =
functors equipped with coherence relations generalizing the relations =
between the operations and cooperations in a bialgebra, so that a bilax =
functor from the one-object monoidal category to VECT would be a =
bialgebra.  This notion, however, only makes sense in the presence of =
braidings on the source and target.

I think Aguiar's usage should prevail, though we also need a name for =
functors between general monoidal categories which are simultaneously =
lax and oplax.

Best Thoughts,
David Yetter=


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Re: bilax_monoidal_functors?=

[Andre Joyal]

Dear David

Thanks for clarifying the notion of Frobenius functor.

In the chapter 4 of the latest version of their book

http://www.math.tamu.edu/~maguiar/a.pdf

Aguiar and Mahajan introduce a notion of P-monoidal functor 
for P is an operad.

If P is the Ass operad (whose models are monoids).
then a P-monoidal functor is a lax monoidal functor,
and if P is the Com operad (whose models are commutative monoids),
then a  P-monoidal functor is a symmetric lax monoidal functor.
Their examples include a notion of Lie-monoidal functor (in the enriched case).
Dually, they introduce a notion of P-comonoidal functor
with the examples of colax (=oplax) monoidal functors
and of symmetric oplax monoidal functors.

But a bilax monoidal functor is not a P-monoidal functor
in the sense of Aguiar and Mahajan because the notion of bialgebra
is defined by a PROP, not by an operad. Similarly a Frobenius monoidal functor
is not a P-monoidal functor because the notion of Frobenius algebra
is defined by a PROP, not by an operad.


The notion of P-monoidal functor for P a PROP 
is not defined in their book.

Any idea?


Best regards,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de David Yetter
Date: ven. 07/05/2010 21:05
À: Categories
Objet : categories: bilax monoidal functors
 
John Baez could not recall whether bilax and Frobenius monoidal functors =
are the same.

The answer is no, in the usage I'd been familiar with,  bilax meant =
simply equipped with both lax and oplax structures, while a Frobenius =
monoidal functor satisfies  additional coherence relation which =
generalize the relations between the multiplication and comultiplication =
in a Frobenius algebra.

A bilax monoidal functor from the one-object monoidal category to VECT =
would be a vector-space with both an algebra and a coalgebra structure =
on it (no coherence relations relating them), while a Frobenius monoidal =
functor would be a Frobenius algebra. =20

Aguiar (with good reason), on the other hand, reserves bilax for =
functors equipped with coherence relations generalizing the relations =
between the operations and cooperations in a bialgebra, so that a bilax =
functor from the one-object monoidal category to VECT would be a =
bialgebra.  This notion, however, only makes sense in the presence of =
braidings on the source and target.

I think Aguiar's usage should prevail, though we also need a name for =
functors between general monoidal categories which are simultaneously =
lax and oplax.

Best Thoughts,
David Yetter=


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Re: bilax_monoidal_functors?=

[Richard Garner]

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Dear André

> In the chapter 4 of the latest version of their book
>
> http://www.math.tamu.edu/~maguiar/a.pdf
>
> Aguiar and Mahajan introduce a notion of P-monoidal functor
> for P is an operad.
>
> ...
>
> The notion of P-monoidal functor for P a PROP
> is not defined in their book.
>
> Any idea?

The notion of P-monoidal functor C --> D, for P an operad, 
may be described most expediently when D has small colimits, 
and these distribute over tensors: for then the functor 
category [C,D] is itself monoidal under Day convolution, and 
a P-monoidal functor is nothing but a P-algebra in this 
functor category.

Such a definition admits an obvious generalisation to the 
case where P is not an operad, but merely a PROP; however, it 
seems to me that such a generalisation has no force. For 
instance, when P is the PROP for coalgebras the notion of 
P-monoidal functor is nothing like that of an oplax monoidal 
functor. The problem is one of variance; in giving a 
comultiplication F -> F * F one is required to map into a 
coend, which is back-to-front.

One way of throwing this into focus is by considering the 
case where D has no colimits to speak of, so that the functor 
category [C,D] is not monoidal, but merely a multicategory. 
In any multicategory, one may still speak of a P-algebra for 
an operad---thereby allowing the notion of P-monoidal 
morphism of Aguiar and Mahajan to find its fully general 
expression---but the notion of P-algebra for an arbitrary 
PROP no longer makes sense.

The moral is that a PROP in general may be built from 
components which originate "in algebra" and components which 
originate "in coalgebra"; or, indeed, from components which 
originate in neither. It is, I think, only when all 
components originate "in algebra"---which is to say that the 
PROP is an operad---that the notion of P-monoidal functor is 
mathematically sensible.

However, all is not lost; for many of the PROPs of interest 
are not just PROPs, but instances of some smaller notion 
which allows "algebraic" and "coalgebraic" components 
interacting according to some particular discipline. For 
example, there is Gan's notion of dioperad, which is 
essentially that of a one-object polycategory. The PROP for a 
Frobenius algebra is an example of such a dioperad. Now, we 
may speak of models for a dioperad in any polycategory; and 
in particular, the functor category [C,D] between two 
monoidal categories bears such a polycategorical structure, 
wherein a model for the dioperad for Frobenius algebras is 
precisely a Frobenius monoidal functor. Jeff Egger has 
studied circumstances under which this polycategorical 
structure of [C,D] is representable, and almost certainly 
discusses this example of Frobenius monoidal functors; but he 
is much more qualified than me to speak on this!

Richard


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