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Stream: theory: category theory

Topic: Yoneda as a right extension?!

Tom Hirschowitz (Nov 23 2022 at 16:12):

For any category $ℂ$, denoting by $𝒫$ the contravariant presheaf endofunctor on $𝐂𝐀𝐓$, we have the restriction functor along Yoneda $𝒫(𝐲_ℂ): 𝒫(𝒫ℂ) → 𝒫ℂ$. Because $ℂ$ is small and Set is complete, this functor has a right adjoint given by right Kan extension. Now if I'm not mistaken, in this particular case, right Kan extension is given precisely by
$𝐲_{𝒫ℂ}: 𝒫ℂ → 𝒫(𝒫ℂ)$. Is this right? Is it standard? Also, my tentative proof is not difficult, but it is set-based, hence the question: does this hold in any Yoneda structure (or variants thereof)?

Nathanael Arkor (Nov 25 2022 at 09:44):

We need to be a little careful about size, but your observation can be deduced both from (1) the theory of lax-idempotent pseudomonads; and (2) from the theory of Yoneda structures (which are closely related structures: see Walker and Di Liberti–Loregian).

(1) The presheaf construction on small categories extends to a pseudomonad $(P, \mu, \eta)$ on the 2-category of locally-small categories exhibiting free small-cocompletion. $P$ is lax-idempotent (as is the case for any pseudomonad expressing some cocompletion). It is a defining property of lax-idempotent pseudomonads that $P \eta_A \dashv \mu_A \dashv \eta_{P A}$. That $\mu_A$ is the same as restriction of presheaves in this example can be seen by exhibiting both as left extensions. (Note that right adjoints are given by left extensions.)

(2) In the context of Yoneda structures, this observation is essentially Corollary 14 of Street–Walters.

Tom Hirschowitz (Nov 25 2022 at 15:10):

Brilliant, thanks!