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Suppose I have a comonad $G$ on a category $\mathbb{C}$.

If $C$ is a preorder then I can define a comonad (i.e. interior operator) $\square$ on the set $\mathcal{P}\uparrow(C)$ of upwards closed subsets of $C$ by taking preimages. This preserves all limits and all colimits and has a right adjoint monad (i.e. a closure operator).

I would like to know if (and how) this can be generalized to the presheaf category $\text{Set}^{C^{\text{op}}}$, since upwards closed sets are generalized by presheafs.

Given a functor $F : C \to C$ one can define a geometric morphism $F : \widehat{C} \to \widehat{C}$ (where $\widehat{C} = \text{Set}^{C^\text{op}}$) where the inverse image functor $F^*$ is given by precomposition by $F^{\text{op}} : C^{\text{op}} \to C^{\text{op}}$ and the direct image functor $F_*$ is given on objects as $F_*(Q)(C) = \text{Hom}\left(\text{Hom}(F(-), C), Q\right)$. Now obviously one gets a comonad on $\widehat{C}$ since $F^* \dashv F_*$, but this does not use the structure of $F$ in any way and doesn't seem to me to be the generalization of the construction on preorders.

My question is, given a comonad $G$ on $C$ is $G^*$ in general a comonad on $\widehat{C}$ and is perhaps $G_*$ a monad right adjoint to it?

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up vote 2 down vote accepted

The operation $\mathbb{C} \mapsto [\mathbb{C}^\mathrm{op}, \mathbf{Set}]$ is a 2-functor $\mathfrak{Cat}^\mathrm{co op} \to \mathfrak{CAT}$, so it must in particular send comonads to monads. For instance, given $\epsilon : G \Rightarrow \mathrm{id}_{\mathbb{C}}$, we get an induced natural transformation $\epsilon^* : \mathrm{id}_{[\mathbb{C}^\mathrm{op}, \mathbf{Set}]} \Rightarrow G^*$ etc.

That said, there is also a 2-functor $\mathfrak{Cat} \to \mathfrak{CAT}$, which sends a functor $G$ to its left Kan extension $G_!$. This doesn't reverse natural transformations, so it sends comonads (resp. monads) on $\mathbb{C}$ to comonads (resp. monads) on $[\mathbb{C}^\mathrm{op}, \mathbf{Set}]$. We can say the same thing for the 2-functor $\mathfrak{Cat} \to \mathfrak{CAT}$ that sends a functor $G$ to its right Kan extension $G_*$.

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This is enough. Upwards closed sets are in fact a subset of covariant functors on a preordered category. So given a comonad on $\mathbb{C}$ we get a monad on $\mathbb{C}^{op}$ which is then mapped to the comonad on $[\mathbb{C}, \mathbf{Set}]$. Thank you. –  Aleš Bizjak Oct 22 '13 at 17:43
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