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Stream: deprecated: our papers

Topic: An introduction to enriched cofunctors

Bryce Clarke (Sep 05 2022 at 09:18):

My coauthor, Matthew Di Meglio, and I have a new preprint out on the arXiv today: An introduction to enriched cofunctors.
Here is the abstract:

Cofunctors are a kind of map between categories which lift morphisms along an object assignment. In this paper, we introduce cofunctors between categories enriched in a distributive monoidal category. We define a double category of enriched categories, enriched functors, and enriched cofunctors, whose horizontal and vertical 2-categories have 2-cells given by enriched natural transformations between functors and cofunctors, respectively. Enriched lenses are defined as a compatible enriched functor and enriched cofunctor pair; weighted lenses, which were introduced by Perrone, are precisely lenses enriched in weighted sets. Several other examples are also studied in detail.

Here is a Twitter thread I wrote on the paper, as well as some talks slides on the topic (here and here).

My paper with Matthew Di Meglio titled "An introduction to enriched cofunctors" is now available on the arXiv: https://arxiv.org/abs/2209.01144 So what are the highlights? (1/n) https://twitter.com/8ryceClarke/status/1566370914035441664

- Bryce Clarke (@8ryceClarke)

Nathanael Arkor (Sep 05 2022 at 11:32):

A quick question: in the paper you state that distributivity of $\mathcal V$ is necessary to define $\mathcal V$-cofunctors, but I noticed that here you suggest distributivity is necessary just to define $\mathcal V\text-\mathbf{Mat}$, from which enriched cofunctors may be extracted. However, if one does not require that composites in $\mathcal V\text-\mathbf{Mat}$ exist (e.g. if one uses a multibicategory or virtual double category), then the distributivity requirement no longer holds. Is there any reason one could not define an enriched cofunctor in this setting instead, and drop the distributivity condition?

@AdrianTMiranda Hi Adrian. Enriched cofunctors *are* a certain kind of monad morphism in the double category of V-matrices. Distributivity is needed to define associativity of composition of V-matrices, which is why is comes up in the definition of enriched cofunctors.

- Bryce Clarke (@8ryceClarke)

Bryce Clarke (Sep 05 2022 at 13:11):

Hi @Nathanael Arkor, thanks for your question. The short answer is: I'm not sure. I have a passing familiarity with virtual double categories, but not really enough experience with them to answer your question. Something I can say is that the enriched cofunctors are a special case of so-called retromorphisms between monads in a double category with companions (defined in Section 5.2 here). So my guess is that one would need to work with something like a virtual equipment to define monad retromorphisms at that level of generality, and then see what you get when specialising to V-Mat.

Bryce Clarke (Sep 05 2022 at 13:17):

I would be very interested in dropping the distributivity condition if it is possible, and I'll definitely think about this some more. If you have any further thoughts, please let me know.

Bryce Clarke (Sep 05 2022 at 13:27):

Screen-Shot-2022-09-05-at-3.21.42-pm.png
Okay, some further thought leads me to believe that a virtual equipment is not sufficient, as I need cells whose target is more that a single arrow. In the screenshot, a monad retromorphism is defined as a (vertical) monad morphism where $f$ is the companion of some horizontal arrow $f \colon A \rightarrow B$. I don't know how to generalise this definition between monoids in a virtual equipment.

Nathanael Arkor (Sep 05 2022 at 14:48):

Ah, I see. I agree it's unclear how to extend $\mathbf{Mnd}$ to virtual double categories (in contrast to the $\mathbf{Mod}$ construction). That explains nicely where the difficulty in dropping distributivity lies, thank you.