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Stream: theory: category theory

Topic: Modules between lax functors in the vtc of double categories

Kevin Carlson (Oct 14 2025 at 17:53):

I was about to DM this to @Nathanael Arkor or @James Deikun but I thought others might be interested to see the answer in public, if you don't mind. In my Topos colloquium talk two weeks ago I mentioned you two's and Hoshino's work on the virtual triple category of double categories. As in the slides here, you describe this vtc as having the three types of 1-cells given by lax functors, tight and loose distributors, and the 2-cells given by (the non-globular generalizations of) tight transformations, loose transformations, and something called alterations. By the way, I had thought I had found a definition of alteration written out somewhere, but perhaps I only saw it in a dream, because I haven't been able to re-Google this.

Kevin Carlson (Oct 14 2025 at 17:54):

Anyway, my real question: as described, the vtc of double categories does not seem to contain the distributors between lax double functors (what Paré and Lambert-Patterson have been calling "modules", after the bicategorical literature.) But Nathanael suggested during my colloquium that actually you can recover these within the triple category, without adding even more kinds of 2-cells. How can this be done?

Nathanael Arkor (Oct 14 2025 at 18:15):

I think the terminology I used on that slide is misleading, and I'm actually not sure why I used it now, sorry! The loose 2-cells specify, for each frame of loose and tight heteromorphisms, a set of "hetero-cells". It's actually the alterations that generalise what I would be inclined to call a "loose transformation" (i.e. what Grandis and Paré call a "vertical transformation").

Nathanael Arkor (Oct 14 2025 at 18:16):

You can see (part of the definition) of an alteration on @Bryce Clarke's talk slides (p. 5) from CT: A new framework for limits in double categories.

Nathanael Arkor (Oct 14 2025 at 18:17):

In particular, when the loose distributors are taken to be identities, we recover the definition of module.

Kevin Carlson (Oct 14 2025 at 18:48):

Ah, yes, Bryce's slides may be where I saw that definition, which would explain why it wasn't Googleable.

Kevin Carlson (Oct 14 2025 at 18:50):

So, say you have a loose 2-cell with loose boundary $1=1,\mathbb{D}=\mathbb{D}$ and tight boundary $F:1\nrightarrow \mathbb{D}, G:1\nrightarrow \mathbb{D}$, that recovers a module between $F$ and $G$ when they're viewed as lax double functors into $\mathbb{S}\mathbf{pan}$?

Nathanael Arkor (Oct 14 2025 at 19:26):

Right. The modules from a lax functor $F \colon \mathbb A \to \mathbb B$ between arbitrary double categories to a lax functor $G$ are obtained most simply as the alterations (i.e. boundary 2-cells) of the following shape:
image.png

However, if you want the modules specifically between lax functors into $\mathbb S\mathbf{pan}$, you can obtain these as loose 2-cells of the following shape:
image.png

Kevin Carlson (Oct 14 2025 at 20:17):

OK, I was still mixing up loose and boundary but I think I'm with you now. The boundary 2-cells, then, generalize both loose transformations and modules, which makes sense as loose transformations are pseudo maps out of $\mathbb{D}\times \mathbb{L}$ while modules are the lax ones (with $\mathbb{L}$ the walking loose). So the loose 2-cells you were describing in terms of their components, are those largely novel?

Nathanael Arkor (Oct 14 2025 at 20:45):

@CB Wells studied a certain class of them (corresponding to the loose 2-cells in the virtual triple category of equipments) in his thesis The Metalanguage of Category Theory as "double profunctors", but as far as I know these are the only place they have appeared before.