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Stream: theory: category theory

Topic: Calculating the Day convolution of Cat-valued presheaves

Jesse Sigal (Aug 09 2021 at 20:20):

Summary of my request for references/help :smile:

Let $(\mathcal{C}, \otimes, e)$ be a monoidal category, and consider the category $[\mathcal{C}^\text{op}, \text{Cat}]$ of $\text{Cat}$-valued presheaves and natural transformations. I want to:

1. Find an specialized definition of Day convolution for $[\mathcal{C}^\text{op}, \text{Cat}]$; and
2. Find an explicit/concrete description of it.

Finding a specialized definition

I'm aware of the standard definition of Day convolution in terms of $\mathcal{V}$-enriched categories. I have also seen the coend formula when $\mathcal{V} = \text{Set}$. However, I do not know any enriched category theory and so cannot at the moment make use of the standard definition.

I also know that $\text{Cat}$-enriched categories are the same as (strict) $2$-categories, but I haven't found any definitions of Day convolution for $2$-categories (which hopefully would be easier to learn than enriched category theory).

In conclusion, I'm looking for a pre-existing specialized definition to $\mathcal{V} = \text{Cat}$ (and thus $\mathcal{C}$ is given the standard/trivial enrichment as $\text{Cat}$ is cocomplete) or a $2$-categorical definition.

Finding an explicit/concrete description

I've successfully applied the definition of Day convolution in the case of $\text{Set}$-valued presheaves. I expanded the coend formula to what it means in terms of universal properties. I also wrote it down as an explicit coequalizer.

In the end I had a description of the resulting set in terms of an equivalence relation generated by the two actions of the morphism on the profunctor I took the coend over.

The above is the level of concreteness I'm looking for.

What I'd like

For (1) I'd appreciate one of

• a link if a definition exists; or
• a quick working out of the general definition in the special case of $\text{Cat}$ if it's trivial for you; or
• telling me I should just invest my time in learning enough enriched category theory to understand the standard definition.

For (2) if someone has a definition, an explicit description of the resulting category (objects and morphisms only is fine).

Thanks in advance for any and all help :smiley:

John Baez (Aug 10 2021 at 01:21):

It sounds like you know how Day convolution works for ordinary Set-valued presheaves on a monoidal category. What sort of descriptions do you know for how to convolve two such presheaves $F, G: \mathcal{C}^{\rm op} \to \mathrm{Set}$? Many of these descriptions generalize painlessly to Cat-valued presheaves. Basically, you just write down the same thing but wherever you wrote "set" you now write "category", and wherever you wrote "function" you now write "functor".

fosco (Aug 10 2021 at 08:08):

You should definitely spend a couple of days learning the definition of enriched category and enriched presheaf, to understand that (so to speak) all categorical notions "transport rigidly" from the $Set$-world to the $\cal V$-world.

What you ask is "trivial" in the following sense: once you have written down the coend

$\displaystyle \int^{XY} FX \times GY \times \hom(\_, X\oplus Y)$

for $Set$-presheaves, interpreted it as a left Kan extension, or as a coequaliser, and found the equivalence relation over $\coprod_{XY} FX \times GY \times \hom(\_, X\oplus Y)$, you do the same things in $Cat$; the only problem is that coequalisers in $Cat$ tend to be quite difficult to describe explicitly even though they always exist.

Just to be sure: your functors ${\cal C} \to {\rm Cat}$ are strict functors, right? Not pseudo.

Jesse Sigal (Aug 10 2021 at 12:18):

@John The one I'm familiar with is the coend version @fosco has written down, I expanded that to what it means in terms of an initial dinatural transformation and also the coequalizer formulation. Thanks for the advice, I'll try the coequalizer version where I now view everything in Cat then :)

Jesse Sigal (Aug 10 2021 at 12:25):

@fosco Thanks for the advice, I'd had a feeling I should probably just sit and learn some enriched category theory :) Hopefully in my case the coequalizers may be simpler since $F$ and $G$ in my case are related and $\mathcal{C}$ is a strict monoidal category.

The functors I have strict, however they could be made to be more lax. I'm currently trying to find some sound semantics for a diagrammatic system I've axiomatized, all of my axioms are phrased in terms of equalities, and this causes the functors to be strict.

Jesse Sigal (Aug 10 2021 at 12:30):

@fosco also, thanks so much for your coend calculus book :smiley:

fosco (Aug 10 2021 at 12:58):

One problem with weaker notions of co/end is that the result of the integral might not be what you expect. For example, the lax coend of the diagram ${\cal C} \to {\sf Cat}$ sending an object to the slice category over it is the twisted arrow category of $\cal C$; and if you laxify the yoneda lemma with coends, what you get is that there is not an isomorphism $F \cong \int^A \hom(\_,A)\times FA$, if $F$ is lax, because the RHS is not lax. Instead, you have pseudo-ified F!

Mike Shulman (Aug 10 2021 at 13:12):

Lax things are weird, but pseudo things are much less weird, and often what you want when doing 2-category theory.

Jesse Sigal (Aug 10 2021 at 13:53):

@fosco Good to know. Funnily enough, the presheaves I want to take the Day convolution of have over categories as their result (but I'm not using the codomain fibration, something similar though). So if I get something weird, I'll be sure to try pseudo versions :)

Jesse Sigal (Aug 10 2021 at 13:54):

@Mike good to know, I'll be sure to keep this in mind if I don't get what I want in my first attempt :)

Jesse Sigal (Aug 10 2021 at 13:55):

I'm actually not convinced that my axioms should even be equalities, it's just that I don't know what my semantics should be yet, so it's good to know if I weaken them to get something pseudo I might make my life better!

Jesse Sigal (Aug 18 2021 at 09:26):

A follow up, last week I learned enough enriched category theory to take coends of $\text{Cat}$-valued functors and thus found the description of Day convolution as a $\text{Set}$-valued map, which was enough for my purposes at the moment.

Jesse Sigal (Aug 18 2021 at 09:27):

Thanks again for all the help and comments :smile:

John Baez (Aug 18 2021 at 17:34):

Great!