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Stream: learning: questions

Topic: self-enrichment as terminal coalgebra

Asad Saeeduddin (May 03 2021 at 20:03):

I'm a bit out of my depth here so I'm not sure exactly how to ask this question, but is there some way to view a self-enriched category as the carrier of a terminal coalgebra of some functor?

Fawzi Hreiki (May 03 2021 at 20:08):

Self enrichment (in the sense of being closed monoidal) is not a universal property to begin with. The same category can be self enriched in different ways.

Jade Master (May 04 2021 at 01:13):

I think there's something about an approach to infinity categories of some sort using this idea but I don't remember the details.

Matt Earnshaw (May 04 2021 at 13:45):

I guess you are thinking of Weak ∞-categories via terminal coalgebras. The starting point is the idea that the category of strict $\infty$-categories is the terminal coalgebra of the endofunctor $\mathcal{V} \mapsto \mathcal{V}\text{-Cat}$

Ben MacAdam (May 04 2021 at 16:43):

I’m pretty sure that for a small monoidal category $V$, you can regard a coalgebra of the comonad $[V,-]$ as a $V$-copresheaf enriched category using the Day convolution tensor product. Maybe the self-enriched one is terminal? There is a very similar result due to Wood that I found in Garner’s “An embedding theorem for tangent categories” (note that they use the monad $V\otimes -$, so there is still a bit of an exercise in translating the 1-categorical result relating the monad and comonad in an SMC induced by a monoid object).

Ben MacAdam (May 04 2021 at 16:45):

I should say, I’m being very sloppy with the word “comonad”, when it should be some sort of 2-comonad.