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Stream: theory: algebraic topology

Topic: Idempotent density comonads, compactly generated spaces

Thomas Read (Aug 30 2021 at 09:11):

Does anyone know non-trivial examples of idempotent density comonads / codensity monads?

For context, I've been thinking about compactly generated spaces, and am interested if there's a nice way to generalise that situation. The story as I understand it goes as follows (realised I was ignoring some size issues but I think this all works still...). Let $\text{Top}$ the category of all topological spaces, $\text{CH}$ the subcategory of compact Hausdorff spaces, $I : \text{CH} \to \text{Top}$ the inclusion functor. Let $k : \text{Top} \to \text{Top}$ the density comonad of $I$ (i.e. the left Kan extension of $I$ along $I$). Then $k$ turns out to be idempotent, so its image is a coreflective subcategory of $\text{Top}$, with coreflector $k$. In fact this is the subcategory of compactly generated spaces, and $k$ is the usual functor replacing a space with a compactly generated space (e.g. as described here).

The surprising part of this story is that the density comonad turns out to be idempotent, so it would be nice to find some general conditions under which this happens. One possible approach is to focus on properties of the one-point space - we can show that a similar thing happens if we replace $\text{CH}$ with any subcategory of $\text{Top}$ containing the one-point space (for example the inclusion of the finite discrete spaces into $\text{Top}$ gives the coreflective subcategory of all discrete spaces). But to really understand this it would help to have examples in categories other than $\text{Top}$.

@Ivan Di Liberti looks like you were thinking about similar things a couple years ago?

Fawzi Hreiki (Aug 30 2021 at 09:45):

I recall reading somewhere that the codensity monad for the inclusion of powers of the Sierpinski space into topological spaces is the soberification monad

Fawzi Hreiki (Aug 30 2021 at 09:46):

The category of algebras is the category of sober spaces (or equivalently, spatial locales) and this monad is idempotent

Thomas Read (Aug 30 2021 at 09:52):

Thanks, that's a good one!

Fawzi Hreiki (Aug 30 2021 at 09:54):

Just a point: every monad is a codensity monad (and likewise dualised) since the relevant Kan extension is just the corresponding adjoint when it exists

Fawzi Hreiki (Aug 30 2021 at 09:55):

So every idempotent monad satisfies your question a priori. But of course, that’s not really that interesting

Thomas Read (Aug 30 2021 at 10:06):

Yeah, I think what I really want are cases where a) the functor you start with isn't already the inclusion of a (co)reflective subcategory, b) the resulting (co)monad isn't the identity

Ivan Di Liberti (Aug 30 2021 at 13:21):

Since I was cited, let me comment a little bit. Back in those days I asked some questions on MO witnessing my interest for codensity monads.
It is in general very hard (almost impossible) to check whether a codensity monad is idempotent, excluding trivial cases. The best I know is the trivial criterion in my MO-question.

All my (relevant) thoughts on the topic have been later condensed in my paper: https://arxiv.org/abs/1910.01014

Thomas Read (Aug 30 2021 at 13:37):

Thanks, the paper looks very interesting

Patrick Nicodemus (Aug 31 2021 at 16:03):

Fawzi Hreiki said:

Just a point: every monad is a codensity monad (and likewise dualised) since the relevant Kan extension is just the corresponding adjoint when it exists

My suggestion for making the conversation interesting would be to replace Kan extension with pointwise Kan extension. I've noticed that, unsurprisingly, these monads have nicer properties than their colleagues.