Q. about monoidal functors

Q. about monoidal functors

[Fred E.J. Linton]

Suppose _A_ is a symmetric monoidal category in the sense
of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_
a monoidal functor.

What, if anything, is known, where τ: X ⊗ Y --> Y ⊗ X
is the symmetry structure on the (symmetric) tensor product ⊗, 
as to whether

[T_X,Y: TX ⊗ TY --> T(X ⊗ Y)] 
and 
[T(τ_X,Y): T(X ⊗ Y) --> T(Y ⊗ X)]

have the same composition as have

[τ_TX,TY: TX ⊗ TY --> TY ⊗ TX]
and
[T_Y,X: TY ⊗ TX --> T(Y ⊗ X)] ?

TIA for any relevant information and/or references thereto.

Cheers, -- Fred





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Re: Q. about monoidal functors

[Steve Lack]

Dear Fred,

Such a T is called a symmetric monoidal functor.

Example: let _A_ be Set with the cartesian monoidal structure. Let
M be a monoid and let T be the functor Set->Set sending X to MxX (which
I'll write as MX). This functor T is monoidal via the map MXMY->MXY sending
(m,x,n,y) to (mn,x,y). It is symmetric monoidal iff M is commutative.

Steve Lack.


On 6/05/10 4:01 PM, "Fred E.J. Linton" <fejlinton@usa.net> wrote:

> Suppose _A_ is a symmetric monoidal category in the sense
> of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_
> a monoidal functor.
>
> What, if anything, is known, where &tau;: X &otimes; Y --> Y &otimes; X
> is the symmetry structure on the (symmetric) tensor product &otimes;,
> as to whether
>
> [T_X,Y: TX &otimes; TY --> T(X &otimes; Y)]
> and
> [T(&tau;_X,Y): T(X &otimes; Y) --> T(Y &otimes; X)]
>
> have the same composition as have
>
> [&tau;_TX,TY: TX &otimes; TY --> TY &otimes; TX]
> and
> [T_Y,X: TY &otimes; TX --> T(Y &otimes; X)] ?
>
> TIA for any relevant information and/or references thereto.
>
> Cheers, -- Fred
>
>
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]



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bilax monoidal functors

[Joyal, André]

Dear All,

In the chapter 3 of their book

"Monoidal functor, species and Hopf algebras"

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

Aguiar and Mahajan introduces 4 kinds of monoidal functors:

1) strong monoidal
2) lax monoidal
3) colax monoidal
4) bilax monoidal

A monoid in a monoidal category C 
is a lax monoidal functor 1-->C, 
a comonoid is a colax monoidal functor 1-->C 
and a bimonoid is a bilax monoidal functor 1-->C.

I wonder who first introduced the notion
of bilax monoidal functor and when?

An example of bilax monoidal functor is
the singuler chain complex functor from 
spaces to chain complexes. The bilax structure
is provided by the Eilenberg-MacLane map
together with the Alexander-Whitney map.

Best,
AJ



-------- Message d'origine--------
De: categories@mta.ca de la part de Steve Lack
Date: jeu. 06/05/2010 19:02
À: Fred E.J. Linton; categories
Objet : Re: categories: Q. about monoidal functors
 
Dear Fred,

Such a T is called a symmetric monoidal functor.

Example: let _A_ be Set with the cartesian monoidal structure. Let
M be a monoid and let T be the functor Set->Set sending X to MxX (which
I'll write as MX). This functor T is monoidal via the map MXMY->MXY sending
(m,x,n,y) to (mn,x,y). It is symmetric monoidal iff M is commutative.

Steve Lack.


On 6/05/10 4:01 PM, "Fred E.J. Linton" <fejlinton@usa.net> wrote:

> Suppose _A_ is a symmetric monoidal category in the sense
> of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_
> a monoidal functor.
>
> What, if anything, is known, where &tau;: X &otimes; Y --> Y &otimes; X
> is the symmetry structure on the (symmetric) tensor product &otimes;,
> as to whether
>
> [T_X,Y: TX &otimes; TY --> T(X &otimes; Y)]
> and
> [T(&tau;_X,Y): T(X &otimes; Y) --> T(Y &otimes; X)]
>
> have the same composition as have
>
> [&tau;_TX,TY: TX &otimes; TY --> TY &otimes; TX]
> and
> [T_Y,X: TY &otimes; TX --> T(Y &otimes; X)] ?
>
> TIA for any relevant information and/or references thereto.
>
> Cheers, -- Fred

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bilax monoidal functors

[John Baez]

André Joyal wrote:


> I wonder who first introduced the notion of bilax monoidal functor and
> when?
>

I believe that Aguiar and Mahajan were the first to formally introduce this
concept, though the Alexander-Whitney-Eilenberg-MacLane example has been
around for a long time.

On the n-Category Cafe, Kathryn Hess recently wrote:

> The A-W/E-Z equivalences for the normalized chains functor are a special
> case of the strong deformation retract of chain complexes that was
> constructed by Eilenberg and MacLane in their 1954 Annals paper "On the
> groups H(π,n). II". For any commutative ring R, they defined chain
> equivalences between the tensor product of the normalized chains on two
> simplicial R-modules and the normalized chains on their levelwise tensor
> product.
>
> Steve Lack and I observed recently that the normalized chains functor is
> actually even Frobenius monoidal. We then discovered that Aguiar and Mahajan
> already had a proof of this fact in their recent monograph. :-)
>
I forget if "Frobenius monoidal" is a precise synonym of "bilax monoidal".

Best,
jb


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Re: bilax monoidal functors

[Andre Joyal]

Dear John,


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?

Best,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de John Baez
Date: ven. 07/05/2010 14:03
À: categories
Objet : categories: bilax monoidal functors
 
André Joyal wrote:


> I wonder who first introduced the notion of bilax monoidal functor and
> when?
>

I believe that Aguiar and Mahajan were the first to formally introduce this
concept, though the Alexander-Whitney-Eilenberg-MacLane example has been
around for a long time.

On the n-Category Cafe, Kathryn Hess recently wrote:

> The A-W/E-Z equivalences for the normalized chains functor are a special
> case of the strong deformation retract of chain complexes that was
> constructed by Eilenberg and MacLane in their 1954 Annals paper "On the
> groups H(?,n). II". For any commutative ring R, they defined chain
> equivalences between the tensor product of the normalized chains on two
> simplicial R-modules and the normalized chains on their levelwise tensor
> product.
>
> Steve Lack and I observed recently that the normalized chains functor is
> actually even Frobenius monoidal. We then discovered that Aguiar and Mahajan
> already had a proof of this fact in their recent monograph. :-)
>
I forget if "Frobenius monoidal" is a precise synonym of "bilax monoidal".

Best,
jb


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Re: bilax monoidal functors

[Michael Batanin]

Dear John and Andre,

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

In this way the numbers in stabilization hypothesis are standarts.
Another advantage, at least for me, is the connection to n-operads.
n-braided higher category is an algebra of an n-braided operad , which 
is a special sort of an n-operad. It is convenient in the
proof of stabilisation hypothesis.

What do you think about this version?

Michael.



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Re: bilax monoidal functors

[Toby Bartels]

Michael Batanin wrote:

>Dear John and Andre,

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

I think that this terminology is dangerous.

While there is a debate to be had over which is better:
John's "k-tuply monoidal" or André's "k-braided monoidal",
the good thing about either of them is that you can understand them
without having to be told precisely how the numbering works,
as long as you follow the rule that "1-foo" = "foo".
So while I also prefer John's numbering to André's,
I'm happy to use either and have had occasion to use both.
But mixing John's numbering with André's terminology would confuse me.

Sometimes the "1-foo" = "foo" rule is violated.
I was very confused when I first saw "n-connected space";
I'd have understood "n-simply connected space" right away.
I tend to think that a lot of established names are badly numbered,
including "n-category" which may be the most basic on a categories list,
but it doesn't really cause any problems as long as "1-foo" = "foo".


--Toby


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Re: bilax monoidal functors

[John Baez]

Michael Batanin wrote:

Dear John and Andre,
>
> I am using a mixture of your terminologies:
>  monoidal = 1-braided
>  braided = 2-braided
>  sylleptic = 3-braided
>

Let's settle this like reasonable people.  I challenge you to a dual.  Name
your choice of weapon!

Best,
jb


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Re: bilax monoidal functors

[Steve Lack]

On 8/05/10 4:03 AM, "John Baez" <john.c.baez@gmail.com> wrote:

> André Joyal wrote:
> 
> 
>> I wonder who first introduced the notion of bilax monoidal functor and
>> when?
>> 
> 
> I believe that Aguiar and Mahajan were the first to formally introduce this
> concept, though the Alexander-Whitney-Eilenberg-MacLane example has been
> around for a long time.

This is also my belief.

> 
> On the n-Category Cafe, Kathryn Hess recently wrote:
> 
>> The A-W/E-Z equivalences for the normalized chains functor are a special
>> case of the strong deformation retract of chain complexes that was
>> constructed by Eilenberg and MacLane in their 1954 Annals paper "On the
>> groups H(¼,n). II". For any commutative ring R, they defined chain
>> equivalences between the tensor product of the normalized chains on two
>> simplicial R-modules and the normalized chains on their levelwise tensor
>> product.
>> 
>> Steve Lack and I observed recently that the normalized chains functor is
>> actually even Frobenius monoidal. We then discovered that Aguiar and Mahajan
>> already had a proof of this fact in their recent monograph. :-)
>> 
> I forget if "Frobenius monoidal" is a precise synonym of "bilax monoidal".
> 

No it's not. Frobenius monoidal is to Frobenius algebras as bilax monoidal
is to bialgebras.

In particular a Frobenius monoidal functor 1-->C is a Frobenius algebra in
C; a bilax monoidal functor 1-->C is a bialgebra in C.

Steve.


> Best,
> jb



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Re: bilax monoidal functors

[John Baez]

John wrote:

> I forget if "Frobenius monoidal" is a precise synonym of "bilax monoidal".
>

Steve replied:


> No it's not. Frobenius monoidal is to Frobenius algebras as bilax monoidal
> is to bialgebras.
>
> In particular a Frobenius monoidal functor 1-->C is a Frobenius algebra in
> C; a bilax monoidal functor 1-->C is a bialgebra in C.
>

Okay, I should have guessed.  So the normalized chains functor from
simplicial abelian groups to chain complexes is both Frobenius monoidal and
bilax monoidal?

We were talking a while back about structures like the group algebra of a
finite group, which is both a Frobenius algebra and a bialgebra.

I guess that means every finite group gives a functor from the terminal
category to Vect that's both Frobenius monoidal and bilax monoidal?   Is
there some slick way to understand why?

Best,
jb


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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

[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

[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

[Michael Shulman]

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

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

Mike


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Re: 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

[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: 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: bilax_monoidal_functors

[Fred E.J. Linton]

Jeff Egger <jeffegger@yahoo.ca> wrote, in part,

> 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.

Without seeking to prolong the use of "autonomous" today,
let me just say in my defense that, at the time I brought
that term into use, I was thinking it was the sort of
place-holder name that would, eventually, be replaced (as
it has been) by something more appropriate. This was, as I
recall, also the original motivation for the term "exact";
fortunately for its coiners, "exact" worked so well that
it never did need to get replaced. "Autonomous," on the 
other hand, was not nearly as felicitous a choice, and has
long since been superceded -- I have no qualms about that,
nor any regrets (all the fewer because, as I recall, I was
at that time thinking only of symmetric closed monoidal
categories V for which the Set-valued Hom functor V(E, -) 
(E the monoidal unit object) was faithful :-) ).

Cheers, -- Fred




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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

[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

[Jeff Egger]

>> 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).

Obviously this is correct.  But, on the other hand, Rel is a 
compact closed category (also: V-Prof, for suitable choice 
of V).  So it is not necessarily the case that every object 
of a compact closed category is small/finite/compact.  

> Indeed, but in this case it is the objects of the category
> which are
> "compact," not the category itself.  So if this is the
> argument, then
> a more natural term would be "locally compact" (clashing
>  with "locally
> small," of course, but agreeing with "locally presentable"
> categories
> in which all objects are presentable).

Hmmm, even that last point is pretty tenuous...  A locally 
presentable category may have the property that every object 
is presentable, but the converse is false.  For example, Sup 
(the category of complete lattices and supremum-preserving 
maps) is not locally presentable; but it is monadic over Set
and therefore has the property in question. 

> (I am *not* proposing to *actually* use "locally compact"
> -- I don't
> want to introduce yet another name for something that
> already has at
> least four names, even if none of the existing four are
> optimal.)

I disagree with this line of argument: if good terminology
can be found, it will kill off its rivals PDQ.  In fact, I 
have not been able to stop myself from thinking about this
issue, and would like to propose "simply closed category" as 
a replacement for "autonomous category" (in the sense of: 
monoidal category in which every object has a left and a 
right dual).  The point is that such a monoidal category is 
(both left and right) closed; moreover, it is one in which 
the "closed structure" (i.e. the pair of internal homs) 
admits an unusually simple description.  

One possible objection, aside from that which Mike has 
already made, is that the word  "simple" already has an 
established mathematical meaning.  My rebuttal to this is 
that there are precedents for using an adverb independently 
of the corresponding adjective.  For example, I see no 
connection between the "completely" in "completely positive 
map" and any of the standard meanings of "complete".  

Cheers,
Jeff.





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