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An article by Kennison

On limit preserving functors, Illinois J. Math. Volume 12, Issue 4 (1968), 616-619.

States the following

Th. Let $C$ be a small category. Then $[C,\text{Set}]_{\Gamma}$ is a reflective subcategory of $[C,\text{Set}]$.

where by $[C,\text{Set}]_{\Gamma}$ he means those functors preserving $\Gamma$-limits for $\Gamma$ a class of cones.

Is it possible to dualize this theorem in the following way:

Conj. Let $C$ be a small category. Then $[C,\text{Set}]^{\Gamma}$ is a coreflective subcategory of $[C,\text{Set}]$.

where by $[C,\text{Set}]^{\Gamma}$ I mean those functors preserving $\Gamma$-colimits for $\Gamma$ a class of cocones.

But is this dualization sound?

In this case $[C,\text{Set}]^{\Gamma}$ would be the category of fixed points of a suitable comonad $T: [C,\text{Set}] \to [C,\text{Set}]$. (A fixed point of $T$ is an object such that $T(X)\cong X$ via the counit of the comonad.)

I am even more interested in knowing if the following is true.

Hope. Is the $[C,\text{Set}]^{\Gamma}$ the category of the fixed points of a pointed (not copointed!) endofunctor $T: [C,\text{Set}] \to [C,\text{Set}]?$

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  • $\begingroup$ I think your conjecture follows from the adjoint functor theorem. It suffices to check that $[C,Set]^\Gamma$ is accessible, and you can write it as a pullback of accessible categories. $\endgroup$ Mar 12, 2018 at 17:46
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    $\begingroup$ That sounds right to me. It's closed under colimits because colimits commute with colimits, and it's accessible since it is the category of models of a (colimit) sketch. $\endgroup$ Mar 12, 2018 at 20:19
  • $\begingroup$ Thanks Mike! That was my argument too. What do you think about the hope? $\endgroup$ Mar 12, 2018 at 20:20

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