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Stream: learning: questions

Topic: Is the endofunctor category on Set duoidal?

Asad Saeeduddin (Apr 04 2020 at 03:16):

I'm trying to figure out whether the endofunctor category on Set is duoidal under the structures of functor composition and Day convolution. I was reading about it on ncatlab, but I'm having a lot of trouble making head or tails of the very abstract explanation given there:

image.png

John Baez (Apr 04 2020 at 03:31):

It's abstract in part because you have to dodge the size issues: $\mathsf{Set}^\mathsf{Set}$ isn't the free cocompletion of $\mathsf{Set}^{\mathrm{op}}$ the way $\mathsf{Set}^{\mathsf{C}}$ is the free cocompletion of $\mathsf{C}^{\mathrm{op}}$ when $\mathsf{C}$ is small, so there's no guarantee Day convolution really exists in the sense of giving $\mathsf{Set}^\mathsf{Set}$ a monoidal structure!

John Baez (Apr 04 2020 at 03:32):

If you have any questions (other than "huh?") please just ask. I don't really know duoidal categories, but now is a good chance for me to learn.

Asad Saeeduddin (Apr 04 2020 at 03:36):

Unfortunately the only question I can muster is "huh?". Is there another category similar to $\mathbf{Set}$ (e.g. that of typed functions perhaps) where I don't have to worry about these subtleties?

Asad Saeeduddin (Apr 04 2020 at 03:50):

Maybe a more sensible way to ask that question is: given a functor category $[A, B]$ that is actually equipped with the monoidal structure of Day convolution, do the functor composition and Day convolution structures together form a duoidal structure on $[A, B]$?

Reid Barton (Apr 04 2020 at 03:54):

How are you going to compose functors if $A \ne B$?

Asad Saeeduddin (Apr 04 2020 at 03:57):

@Reid Barton Good point, my bad. Suppose an endofunctor category $[A, A]$ then.

John Baez (Apr 04 2020 at 06:42):

The comment on nLab says the problem is not unique to $\mathsf{Set}$; it shows up for a very large class of categories $V$. I think it's best to bite the bullet and figure out what they're saying there.

Joachim Kock (Apr 04 2020 at 10:36):

The picture quoted suggests restricting to accessible endofunctors. It may be instructive to restrict even further, considering for example polynomial endofunctors, or endofunctors preserving filtered colimits, or analytic functors (=preserving filtered colimits and weakly preserving cofiltered limits). The latter category is equivalent to the category of species, where the duoidal structure is well known and not difficult to work out by hand -- see for example the book of Aguiar and Mahajan.