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Stream: theory: category theory

Topic: Dual Pitts theorem(s)?

Spencer Breiner (Mar 01 2024 at 16:06):

(One of) Pitts' theorems says that a cosheaf $a:C\to {\bf V}$ is equivalent to a cocontinuous functor $\overline{a}:{\bf Sh}(C)\to{\bf V}$. The proof proceeds by defining a tensor product $s\otimes a$ of sheaves and cosheaves as a coend and setting $\overline{a}=-\otimes a$. Then $a$ preserves covers, so it lifts nicely through the Yoneda embedding. The last is a left Kan extension.

This leaves me with three questions.

• What is the analogous relationship for the dual Yoneda embedding $\tilde{y}:C^{op}\to {\bf CoSh}(C)$.
• Is there an associated universal property for $s\otimes -$?
• Is there an analogous cotensor defined in terms of ends?

Hopefully, at least two of these three will dovetail. Thanks!