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Stream: theory: category theory

Topic: Higher category theory using double categories

Patrick Nicodemus (Feb 18 2024 at 20:04):

This post sketches an algebraic approach to $n$-categories where the higher "transfors" - natural transformations, modifications, and so on - play a more explicit role. I will then ask for pointers to literature on this.

Patrick Nicodemus (Feb 18 2024 at 20:04):

Let $C$ be the category of $\mathbf{Cat}$ graphs, and $T$ the free 2-category monad on $C$. A category is an algebra for this monad; a bicategory is a "pseudoalgebra". A functor is a morphism of algebras; a (pseudo, lax) functor is a (pseudo, lax) morphism of monad algebras. All very nice, but where are the (pseudo) natural transformations and the modifications? (It is easy to define icons in this framework.)

Recently the following occurred to me. $\mathbf{Cat}$ graph is a double category in a natural way, where a "horizontal" morphism is a kind of $\mathbf{Cat}$-profunctor without any interesting coherence conditions. If $(X, A : X\times X\to\mathbf{Cat})$ and $(Y, B: Y\times Y\to\mathbf{Cat})$ are $\mathbf{Cat}$-graphs, we can define a horizontal morphism to be $Z: X\times Y \to \mathbf{Cat}$ together with actions $A(x,x')\times Z(x',y)\to A(x,y)$ and $Z(x,y)\times B(y,y')\to Z(x,y')$; these actions should commute. Similarly, $2Cat$ is a double category where the vertical morphisms are strict 2-functors and the vertical morphisms are $\mathbf{Cat}$-valued profunctors.

Patrick Nicodemus (Feb 18 2024 at 20:04):

More generally if $X$ is any double category there should be a double category of "objects of $X$ equipped with a horizontal endomorphism"; if $X$ is the double category of sets, functions and $\mathbf{Cat}$-matrices we should get this as our category $End(X)$. Similarly there is a double category of "monoids internal to $X$ and modules between them", these are 2-categories in this case.

Patrick Nicodemus (Feb 18 2024 at 20:04):

Now, natural transformations arise explicitly in this framework because they are certain 2-cells in the double category (between identity profunctors). So, this is nice, we can talk about natural transformations now, so we can talk about adjunctions between 2-categories. But still we need modifications, do we need to pass to a bigger formalism beyond double categories? No, we don't! Because our double category has tabulators and cotabulators, and this lets us realize 2-cells as 1-cells satisfying coherence conditions. If $E(C)$ denotes the tabulator of the identity profunctor on the 2-category $C$, then a natural transformation $\tau: F\Rightarrow G$ for $F,G : C\to D$ can be represented as a 2-functor $E(C)\to E(D)$ which commutes with the maps $F, G : C\to D$ and the projections from the tabulator $E(C)\to C$, $E(D)\to D$. Then a modification can be represented as a natural transformation between these functors.

Patrick Nicodemus (Feb 18 2024 at 20:05):

I would speculate that this approach can be extended to higher dimensional category theory. Furthermore, I note that there is no canonical definition of the whiskering of two pseudonatural transformations between pseudofunctors, it involves one ad hoc choice (which in turn forces whiskering to only be associative up to isomorphism) whereas the definition using profunctors gives (I have not checked carefully) a canonical and strictly associative composition.

Patrick Nicodemus (Feb 18 2024 at 20:11):

A weakness of this approach that I should mention is that this definition of "natural transformation" using profunctors does not appear to reasonably agree with the 2-categorical notion of natural transformation between lax functors. If $A,B$ are 2-categories and we adopt this "profunctor based definition", there is no longer a category $Lax(A,B)$ of lax functors and pseudonatural transformations between lax functors, I think - there is no identity pseudonatural transformation on a lax functor in this case, and no associative composition of pseudonatural transformations that I can see.
Imo, this is not a reason to dismiss the approach. Instead, I see it as a different manifestation of the well known problem that it is impossible to whisker a lax functor with a pseudonatural transformation. (In other words, there is no composition functor $Lax(A,B)\times Lax(B,C)\to Lax(A,C)$ in any case, here the obstruction is that those categories are not well-defined to begin with.) Anyway, it should work well when all functors are pseudo, not lax.

Evan Patterson (Feb 18 2024 at 20:44):

Patrick Nicodemus said:

More generally if $X$ is any double category there should be a double category of "objects of $X$ equipped with a horizontal endomorphism"; if $X$ is the double category of sets, functions and $\mathbf{Cat}$-matrices we should get this as our category $End(X)$. Similarly there is a double category of "monoids internal to $X$ and modules between them", these are 2-categories in this case.

Since you ask about pointers to the literature: this sounds like the "Mod" construction, which takes a double category D and produces a virtual double category Mod(D) whose objects are monads/monoids/category objects in D (terminology varies). See, for example, Leinster's book on higher categories or Cruttwell-Shulman on generalized multicategories.

Evan Patterson (Feb 18 2024 at 20:45):

Although Mod(D) is in general virtual, it always has representable units, so it at least has an underlying 2-category.

Evan Patterson (Feb 18 2024 at 20:49):

If you instantiate this construction for D the double category of Cat-matrices (or Cat-graphs, if you prefer), you get the 2-category of 2-categories, 2-functors, and 2-natural transformations, as you say.

Evan Patterson (Feb 18 2024 at 20:54):

Your post also reminds me of my recent paper with Michael Lambert on double theories. One way to think about this work is that it generalizes the Mod construction to theories besides the "trivial one." We show in detail that this machinery generates unital virtual double categories, so in particular 2-categories, of models. But I never really expected to be able to get higher-dimensional cells in the cases where they should exist, such as in Cat-enriched category theory. So your suggestion that tabulators could help do this is very interesting to me.