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In general, given a surjective geometric morphism $f: \mathcal{E} \to \mathcal{T}$ of toposes, then the adjunctions $f^* \vdash f_*$ is comonadic.

In particular, given a topological or localic mononoid $M$, we have a canonical surjective geometric morphism $Set \to B M$ where $BM$ is the topos of sets endowed with a continuous (smooth) action of $M$.

So, there is a comonad on sets whose category of coalgebras is the category of (smooth) $BM$-sets.

In the case where $M$ is a localic group $G$, the underlying functor sends a set $X$ to the set of functions $G \to X$ that have an open stabilizer in the translation action of G ( so a kind of uniform continuity condition I guess). I assume a similar formula works for a monoids, but I'm less familiar with it so I don't want to claim it.

I think one recovers a polynomial example only in the case where the monoid is discrete... This suggest maybe there is a class of comonads corresponding to some topological categories...

In general, given a surjective geometric morphism $f: \mathcal{E} \to \mathcal{T}$ of toposes, then the adjunctions $f^* \vdash f_*$ is comonadic.

In particular, given a topological or localic monoid $M$, we have a canonical surjective geometric morphism $Set \to B M$ where $BM$ is the topos of sets endowed with a continuous (smooth) action of $M$.

So, there is a comonad on sets whose category of coalgebras is the category of (smooth) $BM$-sets.

In the case where $M$ is a localic group $G$, the underlying functor sends a set $X$ to the set of functions $G \to X$ that have an open stabilizer in the translation action of G ( so a kind of uniform continuity condition I guess). I assume a similar formula works for a monoids, but I'm less familiar with it so I don't want to claim it.

I think one recovers a polynomial example only in the case where the monoid is discrete... This suggest maybe there is a class of comonads corresponding to some topological categories...

In general, given a surjective geometric morphism $f: \mathcal{E} \to \mathcal{T}$, then the adjunctions $f^* \vdash f_*$ is comonadic.

In particular, given a topological or localic mononoid $M$, we have a canonical surjective geometric morphism $Set \to B M$ where $BM$ is the topos of sets endowed with a continuous (smooth) action of $M$.

So, there is a comonad on sets whose category of coalgebras is the category of (smooth) $BM$-sets.

In the case where $M$ is a localic group $G$, the underlying functor sends a set $X$ to the set of functions $G \to X$ that have an open stabilizer in the translation action of G ( so a kind of uniform continuity condition I guess). I assume a similar formula works for a monoids, but I'm less familiar with it so I don't want to claim it.

I think one recovers a polynomial example only in the case where the monoid is discrete... This suggest maybe there is a class of comonads corresponding to some topological categories...

In general, given a surjective geometric morphism $f: \mathcal{E} \to \mathcal{T}$ of toposes, then the adjunctions $f^* \vdash f_*$ is comonadic.

In particular, given a topological or localic mononoid $M$, we have a canonical surjective geometric morphism $Set \to B M$ where $BM$ is the topos of sets endowed with a continuous (smooth) action of $M$.

So, there is a comonad on sets whose category of coalgebras is the category of (smooth) $BM$-sets.

In the case where $M$ is a localic group $G$, the underlying functor sends a set $X$ to the set of functions $G \to X$ that have an open stabilizer in the translation action of G ( so a kind of uniform continuity condition I guess). I assume a similar formula works for a monoids, but I'm less familiar with it so I don't want to claim it.

I think one recovers a polynomial example only in the case where the monoid is discrete... This suggest maybe there is a class of comonads corresponding to some topological categories...

In general, given a surjective geometric morphism $f: \mathcal{E} \to \mathcal{T}$, then the adjunctions $f^* \vdash f_*$ is comonadic.

In particular, given a topological or localic mononoid $M$, we have a canonical subjective geometric morphism $Set \to B M$ where $BM$ is the topos of sets endowed with a continuous (smooth) action of $M$.

So, there is a comonad on sets whose category of coalgebras is the category of (smooth) $BM$-sets.

In the case where $M$ is a localic group $G$, the underlying functor sends a set $X$ to the set of functions $G \to X$ that have an open stabilizer in the translation action of G ( so a kind of uniform continuity condition I guess). I assume a similar formula works for a monoids, but I'm less familiar with it so I don't want to claim it.

I think one recovers a polynomial example only in the case where the monoid is discrete... This suggest maybe there is a class of comonads corresponding to some topological categories...

In general, given a surjective geometric morphism $f: \mathcal{E} \to \mathcal{T}$, then the adjunctions $f^* \vdash f_*$ is comonadic.

In particular, given a topological or localic mononoid $M$, we have a canonical surjective geometric morphism $Set \to B M$ where $BM$ is the topos of sets endowed with a continuous (smooth) action of $M$.

So, there is a comonad on sets whose category of coalgebras is the category of (smooth) $BM$-sets.

In the case where $M$ is a localic group $G$, the underlying functor sends a set $X$ to the set of functions $G \to X$ that have an open stabilizer in the translation action of G ( so a kind of uniform continuity condition I guess). I assume a similar formula works for a monoids, but I'm less familiar with it so I don't want to claim it.

I think one recovers a polynomial example only in the case where the monoid is discrete... This suggest maybe there is a class of comonads corresponding to some topological categories...