### 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

How is

related to

?

Here $[X,Y]$ denotes the 2-category of 2-functors from $X$ to $Y$, 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 $[X,Y]$ is an internal hom-object for 2-categories only if we take the tensor product

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

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

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

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

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

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

of 2-functors $F$ and $G$ comes from a collection of squares

I say $\sigma$ is “pseudo” if $\sigma(t)$ 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.

## 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

doget a symmetric monoidal closed category of 2-categories, which everyone callsGray, 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 aGray-category. Another is Nick Gurski’s thesis, available if you’re nice. But, you may want to start with thebriefintroduction toGray-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:

Almost all modern work uses a 2-category called

Gray, defined to consist of:It sounds like this is what you’re really interested in. As you suspect, the underlying category of

Grayis a symmetric monoidal, where the tensor product is defined to be adjoint to the obvious internal hom.The obvious internal hom? Given 2-categories $C$ and $D$, we define $\mathrm{hom}_{\mathbf{Gray}}(C,D)$ to be the 2-category consisting of

You’re probably used to defining an internal hom to be adjoint to the tensor product in a symmetric monoidal category: $\mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, \mathrm{hom}(B,C))$ 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 $\mathbf{Gray}$, give it a ‘Gray tensor product’ $\otimes_{\mathbf{Gray}}$, 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 $\mathbf{Gray}$-category is a category enriched over $\mathbf{Gray}$. 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.