bilax monoidal functors

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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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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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
> 
> 
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


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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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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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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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