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November 21, 2006

Basic Question on Homs in 2-Cat

Posted by Urs Schreiber

I have

John W. Gray
Formal Category Theory: Adjointness for 2-Categories
Springer, 1974

in front of me, but I haven’t absorbed it yet. I am looking for information about the following question:

In the world of strict 2-categories, strict 2-functors, pseudonatural transformations and modifications of these, consider three 2-categories

(1)A,B,C. A, B, C \,.

How is

(2)[A,[B,C]] [A,[B,C]]

related to

(3)[B,[A,C]] [B,[A,C]]

?

Here [X,Y][X,Y] denotes the 2-category of 2-functors from XX to YY, pseudonatural transformations and modifications.

I am interested in this question, because it seems - unless I am hallucinating - to play a role in the construction of extended 2-dimensional quantum field theories #.

I see that one answer to this question is provided by item iii) of theorem I.4.14 of Gray’s text. But I need to better understand what this theorem tells me in practice.

I understand that [X,Y][X,Y] is an internal hom-object for 2-categories only if we take the tensor product

(1)XY X \otimes Y

of two 2-categories to be not the naïve one, but the one defined by Gray in the proof of theorem I.4.9.

This apparently fails to satisfy something like

(2)XYYX. X \otimes Y \simeq Y \otimes X \,.

Of course if such an equivalence existed, it would imply that [A,[B,C]][B,[A,C]][A,[B,C]] \simeq [B,[A,C]].

If I understand correctly, Gray shows is that if we denote by (def I.4.12)

(3)Fun d(A,B) \mathrm{Fun}_d(A,B)

the 2-category of 2-functors, lax-natural transformations and modifications of these, and denote by

(4)Fun u(A,B), \mathrm{Fun}_u(A,B) \,,

the 2-category of 2-functors, op-lax-natural transformations and modifications of these, then (theorem I.4.14, iii))

(5)Fun u(A,Fun d(B,C))Fun d(B,Fun u(A,C)). \mathrm{Fun}_u(A, \mathrm{Fun}_d(B,C)) \simeq \mathrm{Fun}_d(B, \mathrm{Fun}_u(A,C)) \,.

I am interested in pseudonatural transformations, where the 2-cell filling the square in the definition of lax- (“quasi-“) natural transformations is invertible.

Clearly, I should keep reading Gray’s book in more detail and think about this more carefully.

But if anyone feels like providing help, I’d greatly appreciate it.

Maybe to clarify my terminology: a morphism

(6)FσG F \stackrel{\sigma}{\to} G

of 2-functors FF and GG comes from a collection of squares

(7)F(a) F(t) F(b) σ(g) σ(t) σ(b) G(a) G(t) G(b). \array{ F(a) &\stackrel{F(t)}{\to}& F(b) \\ \sigma(g) \downarrow \; &\Downarrow \sigma(t)& \; \downarrow \sigma(b) \\ G(a) &\stackrel{G(t)}{\to}& G(b) } \,.

I say σ\sigma is “pseudo” if σ(t)\sigma(t) here is invertible, I say it is “lax” if it is not necessarily invertible and “op-lax” if it is not necessarily invertible and in fact points in the other direction.

I hope that I am right that what I call “lax” and “op-lax” here is what Gray in his book calls “quasi-d-natural” and “quasi-u-natural”, respectively.

Posted at November 21, 2006 2:46 PM UTC

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22 Comments & 3 Trackbacks

Re: Basic Question on Homs in 2-Cat

It sounds like you want a symmetric monoidal closed category of 2-categories, and are frustrated by Gray’s use of lax natural transformations instead of pseudonatural transformations in defining his internal hom.

If so, you’re in luck: everyone else agrees with you! Nowadays we use pseudonatural transformations, just like you want. So, we do get a symmetric monoidal closed category of 2-categories, which everyone calls Gray, even though it differs slightly from Gray’s original idea.

You might try some more modern references on Gray-categories. One that leaps to mind is Gordon, Power and Street’s book on tricategories, where they prove every tricategory is triequivalent to a Gray-category. Another is Nick Gurski’s thesis, available if you’re nice. But, you may want to start with the brief introduction to Gray-categories in HDA1, starting around page 13.

If you ever need a real expert, try Steve Lack or Ross Street - they’ll be tickled pink to have a real physicist wanting to use Gray. Tom Leinster probably also knows this stuff better than I do.

Gray’s original work was based on the 2-category consisting of:

  • strict 2-categories
  • strict functors between these
  • lax natural transformations between these

Almost all modern work uses a 2-category called Gray, defined to consist of:

  • strict 2-categories
  • strict functors between these
  • pseudonatural transformations between these

It sounds like this is what you’re really interested in. As you suspect, the underlying category of Gray is a symmetric monoidal, where the tensor product is defined to be adjoint to the obvious internal hom.

The obvious internal hom? Given 2-categories CC and DD, we define hom Gray(C,D)\mathrm{hom}_{\mathbf{Gray}}(C,D) to be the 2-category consisting of

  • functors from CC to DD
  • pseudonatural transformations between these
  • modifications between these

You’re probably used to defining an internal hom to be adjoint to the tensor product in a symmetric monoidal category: Hom(AB,C)Hom(A,hom(B,C)) \mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, \mathrm{hom}(B,C)) However, one can also go the other way, and define a tensor product that’s adjoint to a given internal hom. This is Gray’s original approach, and it also works for the new approach outlined above - the only difference is that now people use an internal hom defined using pseudonatural transformations instead of lax natural transformations.

If this seems a bit scary, there’s an equivalent approach where you start with a tensor product and use that to define the internal hom. This is the approach sketched in HDA1. In other words: start with the underlying category of Gray\mathbf{Gray}, give it a ‘Gray tensor product’ Gray\otimes_{\mathbf{Gray}}, get a symmetric monoidal category, and then show it’s closed - i.e., show there’s an internal hom adjoint to this tensor product.

(Gray’s original tensor product involved a noninvertible 2-arrow. Now we use an invertible one.)

A Gray\mathbf{Gray}-category is a category enriched over Gray\mathbf{Gray}. In the expository part of HDA1 we unravel what this means and show that it’s what Kapranov and Voevodsky called a semistrict 3-category. Gordon, Power and Street proved that every tricategory is ‘triequivalent’ to one of these. Nick Gurski’s thesis examines these issues in more detail.

Posted by: John Baez on November 21, 2006 4:39 PM | Permalink | Reply to this

Re: Basic Question on Homs in 2-Cat

Thanks a lot! That was very helpful.

To some extent I was hoping that [A,[B,C]][A,[B,C]] would not be equivalent to [B,[A,C]][B,[A,C]] (for the pseudo case).

Initially I guessed that they should be equivalent (as one would hope). But for the example that I am looking at currently I simply tried to write down what an object of [A,[B,C]][A,[B,C]] could get sent to under this equivalence.

Unfortunately, so far, in that example, I was able to construct an object of [B,[A,C]][B,[A,C]] from one in [A,[B,C]][A,[B,C]] only under restricted condtions. Hence I was beginning to wonder whether it was me or instead the formalism who performed suboptimally. :-)

Thence my question here.

Remarkably, the equivalence

(1)[A,[B,C]][B,[A,C]] [A,[B,C]] \simeq [B,[A,C]]

does not involve the tensor product of 2-categories, so I guess understanding it is to some degree independent from understanding

(2)[A,[B,C]][AB,C] [A,[B,C]] \simeq [A\otimes B,C]

and the intricacies of defining ABA \otimes B (even though, of course, the latter equivalence together with ABBAA\otimes B \simeq B \otimes A implies the former).

Do you know of any reference where a morphism

(3)f:[A,[B,C]][B,[A,C] f : [A,[B,C]] \stackrel{\sim}{\to} [B,[ A,C]

implementing that equivalence is spelled out explicitly?

It should be all obvious, and I must be being dense, but at the moment I am having trouble constructing such an ff in full generality. (But at least I have it now for the special example that I am currently interested in.)

Part of the trouble is of course that what I really need to construct is an explicit realization of the equivalence

(4)[A,[B,[C,D]]][C,[A,[B,D]]]. [A,[B,[C,D]]] \simeq [C,[A,[B,D]]] \,.

Related to that:

do you have a nice way to think about how to construct

(5)A GrayB A \otimes_\mathbf{Gray} B

?

I have looked at Gray’s description and didn’t feel particularly enlightened afterwards.

So I understand that objects of A GrayBA \otimes_\mathbf{Gray} B are simply pairs (a,b)(a,b) of objects of AA and BB, respectively.

Next, morphisms are certain equivalence classes of composable sequences of morphisms in the naive A×BA\times B. One throws in an invertible 2-cell

(6)γ f,g:(f,Id)(Id,g)(Id,g)(f,Id) \gamma_{f,g} : (f,\mathrm{Id})(\mathrm{Id},g) \stackrel{\sim}{\to} (\mathrm{Id},g)(f,\mathrm{Id})

for every pair (f,g)(f,g) in the naive Hom(A×B)\mathrm{Hom}(A\times B) and demands some compatibilities with the existing 2-morphisms of AA and BB.

Is there an easy way to make this more precise?

Posted by: urs on November 21, 2006 5:42 PM | Permalink | Reply to this

Re: Basic Question on Homs in 2-Cat

Urs wrote:

Remarkably, the equivalence

(1)[A,[B,C]][B,[A,C]] [A,[B,C]] \simeq [B,[A,C]]

does not involve the tensor product of 2-categories, so I guess understanding it is to some degree independent from understanding

(2)[A,[B,C]][AB,C] [A,[B,C]] \simeq [A\otimes B,C]

and the intricacies of defining ABA \otimes B.

Well, there’s nothing terribly intricate about defining ABA \otimes B if we’re talking about the Gray tensor product - Gordon, Power and Street do it in their paper. I would tell you myself, but I don’t have time to draw the necessary diagrams here! You can figure them out by looking at Lemma 4 of HDA1.

Since the Gray tensor product is really not so bad, I prefer to tackle your question about the internal hom using the symmetry of this tensor product. For example:

(3)[A,[B,C]][AB,C][BA,C][B,[A,C]]. [A,[B,C]] \simeq [A \otimes B, C] \simeq [B \otimes A, C] \simeq [B,[A,C]] .

But, you’re right: it’s possible to study closed categories without even mentioning the tensor product, and that’s how Eilenberg and Kelly originally did it! In their foundational paper:

  • Samuel Eilenberg and G. Max Kelly, Closed categories, in Proceedings, Conference on Categorical Algebra, La Jolla, 1965 Springer-Verlag, New York, 1966, pp. 421-562.

they introduced a kind of “closed category” that wasn’t monoidal, including some coherence laws - a bit like mutant versions of the pentagon identity - that seem quite sneaky to those of us raised on monoidal categories.

It’s just a property for such a closed category to admit a tensor product adjoint to the internal hom:

(4)Hom(AB,C)Hom(A,[B,C]) Hom(A \otimes B, C) \simeq Hom(A, [B,C])

If such a \otimes exists, it’s unique up to natural isomorphism, and one gets a monoidal category. One also gets the internalized version of the above adjunction:

(5)[AB,C][A,[B,C]]. [A \otimes B, C] \simeq [A, [B,C]].

John Gray took this viewpoint in his original work on the Gray tensor product.

In modern work, we usually reverse the logic and start by defining the Gray tensor product of 2-categories. We get a symmetric monoidal category called Gray\mathbf{Gray}.

It’s just a property for a monoidal category to admit an internal hom adjoint to the tensor product:

(6)Hom(AB,C)Hom(A,[B,C]) Hom(A \otimes B, C) \simeq Hom(A, [B,C])

If such a [,][-,-] exists, it’s unique up to natural isomorphism, and one gets a closed category. One also gets the internalized version of the above adjunction:

(7)[AB,C][A,[B,C]] [A \otimes B, C] \simeq [A, [B,C]]

So, it’s all nice and symmetrical - being a closed monoidal category is a property of a closed category, but also a property of a monoidal category.

I urge you to read some more modern stuff on the symmetric monoidal category Gray\mathbf{Gray}, to complement what Gray himself did.

Posted by: John Baez on November 22, 2006 4:41 PM | Permalink | Reply to this

Re: Basic Question on Homs in 2-Cat

they’ll be tickled pink to have a real physicist wanting to use Gray.

In fact, that “real physicist” really wants to have

(1)[A,[B,C]][B,[A,C]] [A,[B,C]] \simeq [B,[A,C]]

not just for 2-categories, but for 3-categories (because that should apply to the case of Chern-Simons theory).

I am getting a little sceptical, even if such an equivalence of 3-functor 3-categories exists, that I will be able to make effective use of it. But it would be cool at least to know that it exists.

Does it? (for suitable definitions of 3-functor 3-categories?)

Posted by: urs on November 21, 2006 5:54 PM | Permalink | Reply to this

Re: Basic Question on Homs in 2-Cat

It should be useful to point out the monoidal closed structure on globular omega-categories which has been explored by Crans and by Street is also developed in

116. (with F.A. AL-AGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach’, Advances in Mathematics, 170 (2002) 71-118.

because the structure in the cubical case is quite transparent: indeed it was so developed for the groupoid case in
48. (with P.J. HIGGINS), “Tensor products and homotopies for $\omega$-groupoids and crossed complexes”, {\em J. Pure Appl. Alg.} 47 (1987) 1-33.

and used to get the monoidal closed structure for crossed complexes, which has the advantage of being able to be written down quite explicitly, and gives homotopies and higher homotopies. Also the structure is symmetric in the groupoid case. An explicit formulation is more difficult for the category case.

One motivation for all this was the idea of higher dimensional nonabelian local-to-global theorems. For this we needed gadgets which could express `algebraic inverse to subdivision’, hence the convenience of cubes, and then `commutative cubes’, hence the connection structure. I remind people that the terms `connection’ and the `transport law’ were borrowed from Virsik’s formulation of path connections, and is much more general than used in the above.


The commutative cube idea is developed in

P.J. Higgins, `Thin elements and commutative shells in cubical
omega-categories’, TAC 40 (2005) 60-74.

R. Steiner, `Thin fillers in the cubical nerves of omega-categories’, TAC 16 (2006) 144-173.

The cubical theory is very geometric. It would be nice to think that the geometric elegance should be reflected in physical applications! But the relation with the globular case is essential to develop the key notions of thin elements, and their composition.

I have not gone into showing an explicit relation with the Gray tensor product, etc..

Ronnie

24 Nov 2006

Posted by: Ronnie Brown on November 24, 2006 8:29 PM | Permalink | Reply to this

Re: Basic Question on Homs in 2-Cat

Some of this discussion on “pseudo”, “lax”, and “op-lax” actually seems to be related to a question Zuckerman and I had, which we can’t seem to get a straight answer from anyone on while Kapranov is out of town and Gurski is in hiding.

We’ve got a “functor” (actually a few examples of them) that doesn’t preserve compositions up to a natural isomorphism, but only up to a natural transformation.

For example, for the Zuckerman functor we have an inclusion Γ(X)Γ(Y)Γ(XY)\Gamma(X)\otimes\Gamma(Y)\rightarrow\Gamma(X\otimes Y), but not in general in the other direction. On the other hand, in his search for an adjoint to the power set functor, the functor U on the category of relations which takes the union of the elements of a set (which are themselves sets) satisfies U(rs)U(r)U(s)U(r\circ s)\rightarrow U(r)\circ U(s), and that’s a proper inclusion.

So, is there a name for this weaker sort of “functor”? Our apologies if this is trivial, but right now we’re the only ones around the department who really think about 2-categories much.

Posted by: John Armstrong on November 21, 2006 5:53 PM | Permalink | Reply to this

lax functors

So, is there a name for this weaker sort of “functor”?

I don’t know what the “Zuckerman functor” is, but I can say what a lax functor is.

A lax functor

(1)F:AB F : A \to B

from a 1-category AA to a 2-category BB is

a map from morphisms in AA to 1-morphism in BB,

for each object aa of AA a 2-morphism

(2)Id F(a)F(Id a) \mathrm{Id}_{F(a)} \Rightarrow F(\mathrm{Id}_a)

(the “unitor”)

and for each pair of morphisms f,gf,g in AA a 2-morphism

(3)F(f)F(g)F(fg) F(f)\circ F(g) \Rightarrow F(f\circ g)

(the “compositor”)

such that compositor and unitor satisfy coherence equations that express the idea of an associative product with unit.

I guess you knew that much yourself. But maybe somebody else reading this did not.

Posted by: urs on November 21, 2006 6:05 PM | Permalink | Reply to this

Re: lax functors

I guess you knew that much yourself. But maybe somebody else reading this did not.

Actually, I didn’t. This is what happens when nobody around talks about a subject: the people who are interested end up having these huge gaps in their knowledge. This blog is helping, but still a lot of my knowledge of higher categories is spackled with my intutition of how things probably work.

Anyhow, is this definition also in Gray? Any recommendations of other basic texts on these sorts of things while we’re at it?

Posted by: John Armstrong on November 21, 2006 6:39 PM | Permalink | Reply to this

Re: lax functors

One should really be thinking of the 1-category AA here as a 2- (bi-)category with only identity 2-morphisms. Then this reduces to a special case of a lax 2-functor.

is this definition also in Gray?

It seems that in that book one only sees strict 2-functors.

Posted by: urs on November 21, 2006 6:53 PM | Permalink | Reply to this

Re: lax functors

John wrote:

Any recommendations of other basic texts on these things while we’re at it?

If you’re after a bare-bones presentation of basic terminology, you could try my Basic Bicategories.

In my youthful naivety, I stuck faithfully to the established terminology, which is disastrous. Nowadays there’s a move towards making the terminology uniform: use strict when things (such as tensor products) are really genuinely preserved, in the sense of being equal; use weak or pseudo when they are preserved up to isomorphism (or the appropriate notion of equivalence); and use lax when they are only preserved up to a connecting map. There’s also colax for when the connecting map goes in the other direction. ‘Weak’ and ‘pseudo’ seem to be fighting it out for the middle notion; personally, I prefer ‘weak’.

Another paper giving the basic definitions of bicategory theory is

Ross Street, Categorical structures, Handbook of Algebra Volume 1 (editor M. Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0 444 82212 7) 529-577.

There’s a lot more in there besides; it’s an interesting read.

Posted by: Tom Leinster on November 21, 2006 8:41 PM | Permalink | Reply to this

Re: lax functors

Basic bicategories is great - its where I learnt the definitions of weak 2-functors, transformations and modifications.

The section on bicategories from Borceux’s `Handbook of Categorical Algebra 1’ is also a nice reference.

Posted by: Bruce Bartlett on November 21, 2006 8:53 PM | Permalink | Reply to this

Re: lax functors

Basic bicategories

For the record, I should maybe mention that what I called the “compositor” and the “unitor” above # correspond, respectively, to φ ABC\varphi_{ABC} and φ A\varphi_{A} on the top of p. 4 in Basic Bicategories.

What I called “coherence equations that express the idea of an associative product with unit” are the two axioms below that.

Posted by: urs on November 22, 2006 9:43 AM | Permalink | Reply to this

explicit transformations

Basic Bicategories

For actually working with explicit bicategories and their morphisms, I find it helpful to “evaluate” natural transformation like the φ ABC\varphi_{ABC} on top of p.4 in terms of the 2-morphisms in the bicategory that are defined by them - like the φ gf\varphi_{gf} appearing on p.4 in the line “thus 2-cells φ gf\varphi_{gf} […]”.

For the special case of strict 2- and 3-categories and their transformations, I have tried to write down the corresponding diagrams in my notes Morphisms of 3-Functors, which I posted here a while ago.

Even though I don’t spell it out there, it is rather straightforward to extend these diagrams to the non-strict case: for instance the compositor corresponds to a certain filled triangle in this description, the associator to a filled hexagon, and so on; and it is always obvious from the context where we have to include these.

I had become fond of these pseudonatural transformations of 2-functors after I realized that a couple of seemingly opaque constructions that people considered in the context of gerbes with connection begin to look perfectly natural once one realizes that they are nothing but examples of the diagrams mentioned above. For instance the funny twist on the connection 1-form of a nonabelian principal bundle gerbe is explained this way, as described here, by realizing it as coming from a pseudonatural transformation.

And this is really just one aspect of how the entire cocylce data of a nonabelian gerbe with connection is just a realization of pseudonatural transformations and modifications of 2-functors with values in a strict 2-group (as described here, with the (prettified) computations being here).

Or the notion of morphism of a bundle gerbe: people first had that defined in the wrong way, and only later realized that the right notion of morphism of bundle gerbe involves some sort of “twist”. Unfortunately, for this historical reason the correct notion of isomorphism of bundle gerbes is therefore now known as a “stable isomorphism”.

This accident would not have happened had people known how a bundle gerbe is a 2-functor and how a “stable isomorphism” of bundle gerbes is a pseudonatural transformation of that 2-functor: the diagrams mentioned above then tell you precisely how that “twist” has to look like. I describe this at the end of these notes (and with more details, probably less readable, here).

Posted by: urs on November 22, 2006 10:08 AM | Permalink | Reply to this

Re: lax functors

Tom points out that thre’s a need for uniform terminology. Strict - that’s fine.
Weak or pseudo - let’s choose one and enshrine it. But Tom doesn’t mention coherence of the isomorphisms. Ditto
for the `connecting map’ in lax - by which
I hope he means e.g from F(f)F(g) to F(fg)
- but how should I remember which direction is which?

But now, how should we decorate the words when we pass to n-cats? Strict is still fine, but the coherence I refer to above
would in that context mean equality,
whereas at level 2, the coherence itself would be up to ?modiification? with
the modifications being strictly coherent?

I’m happy in the limit with the terminology of (strongly) homotopy coherent functor.

Posted by: jim stasheff on November 23, 2006 3:46 PM | Permalink | Reply to this

Re: lax functors

Jim Stasheff wrote:

Tom doesn’t mention coherence of the isomorphisms

Right, I should have done. I was trying to put it briefly, and took it as read that everything should be coherent.

Jim also wrote:

Ditto for the `connecting map’ in lax - by which I hope he means e.g from F(f)F(g) to F(fg) - but how should I remember which direction is which?

I remember it like this. In a lax functor, the connecting map is

(1)F(f)F(g)F(fg), F(f) \circ F(g) \rightarrow F(f \circ g),

going from the composite of two things to one thing. This is just like the multiplication in a monoid,

(2)M×MM, M \times M \rightarrow M,

going from the product of two things to one thing. If all the arrows are reversed, we get colax functors and comonoids.

So a lax functor is like a monoid, and a colax functor is like a comonoid.

In fact, you can make this perfectly precise. Let C\mathbf{C} be a monoidal category, and denote by 1\mathbf{1} the terminal monoidal category. Then a lax monoidal functor 1C\mathbf{1} \rightarrow \mathbf{C} is exactly a monoid in C\mathbf{C}, and dually for colax and comonoids.

Jim:

how should we decorate the words when we pass to n-cats?

I don’t know. What if we have a functor between n-categories that preserves the structure not strictly, or up to isomorphism, but at the next level of equivalence? Perhaps this is a “2-weak n-functor”? Then there are similar questions for transformations, adjunctions, monads, algebras for monads, etc.

If we simply speak of a “weak” thing, that should mean “weak in the most generous possible sense”.

And when you allow things to be lax, there are further questions of orientation. Enumerating the possibilities, and discovering the structure formed by the set of all possibilities, looks like a serious task in itself.

Posted by: Tom Leinster on November 23, 2006 4:13 PM | Permalink | Reply to this

Re: lax functors

Let CC be a monoidal category, and denote by 1\mathbf{1} the terminal monoidal category. Then a lax monoidal functor 1C\mathbf{1}\to C is exactly a monoid in CC.

I like to think of it this way, suspending everything monoidal in the picture:

Let DD be a 2-category. Let 11 be the trivial 1-category on a single object.

Then a lax functor

(1)1D 1 \to D

is precisely a monad in DD.

The above example follows for D=Σ(C)D = \Sigma(C), with CC monoidal.

What I like about this “suspended” point of view is that it seems to nicely generalize to interesting examples.

For instance let

(2)p={ab} p = \{a \to b\}

be the category with two objects and a single nontrivial morphism, as shown. Then a lax functor

(3)pΣ(C) p \to \Sigma(C)

is

an algebra AA in CC (I think of CC being abelian here, hence I say “algebra” instead of “monoid”)

together with an algebra BB in CC

together with an AA-BB bimodule in CC.

Let, more generally, pp be any category. Then a strict functor

(4)pBim(C) p \to \mathrm{Bim}(C)

is a lax functor

(5)pΣ(C). p \to \Sigma(C) \,.

(I’d hope also the converse were true, but I am not sure if or how the lax functor would enforce the universality of the tensor product of bimodules.)

Another application might be enriched categories.

Let AA be a CC-enriched category with object set Obj(A)\mathrm{Obj}(A).

Then I think this is the same as a lax functor from the pair groupoid of Obj(A)\mathrm{Obj}(A) to Σ(C)\Sigma(C).

I once made some remarks on how that point of view might be relevant for CFT here.

Posted by: urs on November 23, 2006 4:32 PM | Permalink | Reply to this

Re: lax functors

Urs wrote (concerning a monoidal category CC):

Another application might be enriched categories.

Let A be a C-enriched category with object set Obj(A).

Then I think this is the same as a lax functor from the pair groupoid of Obj(A) to Σ(C).

Yes!

(For other readers: the pair groupoid of a set SS is otherwise known as the indiscrete or codiscrete category on SS. Its object-set is SS and each hom-set has exactly one element. This implies that every map is an isomorphism.)

You can take this point of view on enriched categories and use it to generalize the notion of enrichment. This allows you to talk about categories enriched in/over multicategories, multicategories enriched in 2-multicategories, …, n-multicategories enriched in (n+1)(n + 1)-multicategories, … , and more besides.

I won’t explain here what I mean by “nn-multicategory”… but it’s something opetopic. This has been worked out in some detail; there’s a two-page summary on p.174-6 of my book.

Posted by: Tom Leinster on November 23, 2006 5:09 PM | Permalink | Reply to this

Re: lax functors

Let AA be a CC-enriched category with object set Obj(A)\mathrm{Obj}(A).

Then I think this is the same as a lax functor from the pair groupoid of Obj(A)\mathrm{Obj}(A) to Σ(C)\Sigma(C).

Yes!

I once thought about how morphisms of lax functors would correspond to morphisms of enriched categories. I don’t recall which conclusion I reached. But something looked funny.

Let’s see. A morphism

(1)k:FG k : F \to G

between lax functors FF and GG to Σ(C)\Sigma(C) would involve 2-cells in Σ(C)\Sigma(C) of the form

(2) F(a,b) k a k a,b k b G(a,b) . \array{ \bullet &\stackrel{F(a,b)}{\to}& \bullet \\ k_a \downarrow \;\; &\Downarrow k_{a,b}& \;\; \downarrow k_b \\ \bullet &\stackrel{G(a,b)}{\to}& \bullet } \,.

But a morphism

(3)k:RS k : R \to S

of enriched categories (that is the identity on objects) has at this point just a 2-cell with vertical morphisms being identities

(4) Hom R(a,b) Id k a,b Id Hom S(a,b) . \array{ \bullet &\stackrel{\mathrm{Hom}_R(a,b)}{\to}& \bullet \\ \mathrm{Id} \downarrow \;\; &\Downarrow k_{a,b}& \;\; \downarrow \mathrm{Id} \\ \bullet &\stackrel{\mathrm{Hom}_S(a,b)}{\to}& \bullet } \,.

Something like this was irritating me. But I realize I would have to think this through again.

Posted by: urs on November 23, 2006 5:31 PM | Permalink | Reply to this

Re: lax functors

By the way, maybe you can help me understand the following.

Somehow an enriched category, then, is something like the “graph” of a lax functor (here “graph” is meant in the sense of - or rather analogous to - the concept of the graph of a function).

Recently, I am trying to understand what this is trying to tell me in the following example:

There is that funny 2-group which I call String G\mathrm{String}_G. Disregarding it’s smooth structure, this is an “extension” of the indicrete category of the set of based paths in the group GG.

The point is that there is an interesting lax functor from that indiscrete category to Σ(U(1)Tor)\Sigma(U(1)\mathrm{Tor}).

The way Alissa, Danny, John and myself conceived this situation was by saying that we get a groupoid whose Hom\mathrm{Hom}-sets are U(1)U(1)-torsors.

So, from the context that we are talking about, we took that lax functor and regarded it as a U(1)TorU(1)\mathrm{Tor}-enriched category - which happens to be a strict 2-group in this case.

However, from the point of view of the lax functor, the following step is natural, while from the point of view of the 2-group it is not:

By simply associating a complex 1-d vector space via the canonical rep of U(1)U(1), we can turn our lax functor to Σ(U(1)Tor)\Sigma(U(1)\mathrm{Tor}) into a lax functor into Σ(Vect )\Sigma(\mathrm{Vect}_\mathbb{C}).

Conversely, this can now be thought of as a Vect \mathrm{Vect}_\mathbb{C}-enriched category - but no longer as a 2-group.

There is something deep here about how the String 2-group, the canonical U(1)U(1)-bundle gerbe on GG and the canonical line bundle gerbe of GG are all different aspects of the same thing.

And underlying this is the correspondence between lax functors and enriched categories.

Well, so far this may not sound like it’s that deep or mysterious. But consider this:

There is a monoidal structure on the indiscrete category of the set of based paths in GG. So we might rather want to think of the entire situation here not in terms of lax functors whose domain is a 1-category - but in terms lax functors whose domain is a 2-category.

I guess these should be called “lax 2-functors”?

Anyway, I think to really appreciate the nature of that String 2-group, and its interdependencies, we would need to understand how it really comes from a lax 2-functor with values in Σ(Σ(Vect ))\Sigma(\Sigma(\mathrm{Vect}_\mathbb{C})).

(For those readers who know what I am getting it: I want to understand how that structure comes not from a 1-gerbe on GG but from a 2-gerbe on BGB G.)

So, er, what is actually my question here… I guess this:

Is there an established theory that defines and relates the concepts that I would call lax 2-functors and enriched 2-categories, analogous to how ordinary lax functors are related to ordinary enriched categories?

Posted by: urs on November 23, 2006 6:19 PM | Permalink | Reply to this

Re: lax functors

I guess you knew that much yourself. But maybe somebody else reading this did not.

Actually, I didn’t. This is what happens when nobody around talks about a subject

By the way, maybe you did see this before, only that it wasn’t called this way.

If you know what a stack is, you implicitly know what a lax functor is.

Like a sheaf (on a site SS, a 1-category) is a functor

(1)SSet S \to \mathrm{Set}

(satisfying some gluing condition) a stack is a pseudofunctor

(2)SCat S \to \mathrm{Cat}

from the 1-category SS to the 2-category Cat\mathrm{Cat} (satisfying some gluing condition).

A pseudofunctor is just like a lax functor, only that the “compositor” and the “unitor” are required to be invertible.

For instance Moerdijk in Introduction to the language of stacks and gerbes defines a stack without mentioning (def. 2.1) the word “pseudofunctor”. In his notation the “compositor” is called τ\tau and the coherence law which says that it is “associative” is the commuting diagram he gives in the middle of p. 7.

The entire definition 2.1 here just says “a fibered category is a pseudofunctor to Cat\mathrm{Cat}”, which in turn just says: “a stack is a lax functor with invertible compositor and unitor, that satisfies some gluing condition”.

Posted by: urs on November 22, 2006 11:16 AM | Permalink | Reply to this

Re: Basic Question on Homs in 2-Cat

The last couple days have found Dr. Stasheff and myself having a conversation about this subject, the conclusion of which he suggested I share with the whole group.

Basically, he was asking more about these “Zuckerman functors”. Dr. Zuckerman did manage to supply an axiomatic definition yesterday:

Let A be a monoidal abelian category. Consider a torsion class (see S. Dickson) which is closed under the monoidal product. Let T be the corresponding torsion functor. Then for any pair of object U and V, we have an inclusion TU times TV into T(U times V). It is interesting to know when this is an equality.

Suppose g is a Lie algebra and k is Lie subalgebra. Then consider the class of all g-modules M such that M is a union of finite dimensional k-submodules. We obtain a torsion class closed under tensor product of g-modules. The corresponding torsion functor “Gamma” is commonly known as the Zuckerman functor in the representation theory literature. (See Enright and Wallach, Duke, 1980.)

So, this is what I now know is called a “lax functor” if we consider tensor product as 1-composition. Maybe “lax monoidal”? It still seems arbitrary to pick the directions of lax and oplax, but I suppose a lot is arbitrary when you really think about it.

Anyhow, Stasheff raised the question about coherence of these maps. I don’t know the answer for the general case, but for Lie algebra representations Γ\Gamma picks out a nice concrete submodule, so I’d be surprised if it weren’t coherent. Still, it is something to check.

My explanation for the issue not being raised in class, which is what it was suggested I repeat on the board, is as follows:

The thing is, it’s even less likely that a bunch of hard-nosed Lie algebra representation people (as the class was, by and large) are going to raise the issue of coherence for a functor preserving a tensor product than for a lax functor, and it’s not very likely there either. “Preserves tensor product” is nice because it helps calculate, but there are lots of people out there who just don’t think about things like coherence. For that matter, there are algebraists (not just analysts or applied mathematicians) who have told me that they would rather not have categories around at all.

I think those of us who even think about higher categories have the freedom to do so from being in relatively cosmopolitan pockets of mathematics, but there is a “flyover country” for math where they don’t like our coherence-sipping categorical kind.

I should also add that the reason I didn’t bring it up was that I was only a second-year student at the time and hadn’t thought about coherence beyond seeing the Coherence Theorem for monoidal categories.

Posted by: John Armstrong on November 25, 2006 6:13 PM | Permalink | Reply to this
Read the post The Baby Version of Freed-Hopkins-Teleman
Weblog: The n-Category Café
Excerpt: The Freed-Hopkins-Teleman result and its baby version for finite groups, as explained by Simon Willerton.
Tracked: November 23, 2006 8:39 PM

Re: Basic Question on Homs in 2-Cat

Given the Gray tensor product G\otimes_G, it seems I can still form the obvious projections A GBp 1A A \otimes_G B \stackrel{p_1}{\to} A and A GBp 2B. A \otimes_G B \stackrel{p_2}{\to} B \,.

These would act in the obvious way on the generators {(Id,f)|fMor(B)}\{(\mathrm{Id},f) | f \in \mathrm{Mor}(B)\} and {(g,Id)|gMor(A)}\{(g,\mathrm{Id}) | g \in \mathrm{Mor}(A)\}, and would send the 2-morphisms (Id,f)(g,Id)(g,Id)(Id,f) (\mathrm{Id},f)\circ (g,\mathrm{Id}) \stackrel{\sim}{\Rightarrow} (g,\mathrm{Id})\circ(\mathrm{Id},f) to the identity 2-morphisms on gg or on ff, respectively.

Is that right?

(Of course I am asking due to my considerations here.)

Posted by: urs on January 26, 2007 10:26 AM | Permalink | Reply to this
Read the post Globular Extended QFT of the Charged n-Particle: String on BG
Weblog: The n-Category Café
Excerpt: The string on the classifying space of a strict 2-group.
Tracked: January 26, 2007 2:57 PM
Read the post QFT of Charged n-Particle: Extended Worldvolumes
Weblog: The n-Category Café
Excerpt: Passing from locally to globally refined extended QFTs by means of the adjointness property of the Gray tensor product of the n-particle with the timeline.
Tracked: August 2, 2007 7:37 PM

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