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Stream: learning: reading & references

Topic: Monadicity of presheaf restriction to right class of OFS

Amar Hadzihasanovic (Jun 08 2025 at 15:19):

The following fact is quite easy to prove: if $C$ is a category with a strict factorisation system $(\mathcal{L}, \mathcal{R})$, and $C_\mathcal{R}$ is the wide subcategory on $\mathcal{R}$-morphisms, then the presheaf restriction functor $\mathrm{PSh}(C) \to \mathrm{PSh}(C_\mathcal{R})$ is monadic.
One can explicitly describe the monad on $\mathrm{PSh}(C_\mathcal{R})$ as sending a presheaf $X$ to the presheaf whose $c$-elements are pairs $(\ell\colon c \to d, x \in X(d))$ where $\ell$ is a $\mathcal{L}$-morphism; one uses the unique factorisation to define the action of $\mathcal{R}$-morphisms on these elements.
Does anyone know an explicit reference for this fact?

Nathanael Arkor (Jun 08 2025 at 17:19):

This is an instance of the fact that restriction along [[dominant functors]] (and, in particular, identity-on-object functors) is monadic. E.g. a reference is Example A4.2.7(b) of the Elephant.

Nathanael Arkor (Jun 08 2025 at 17:20):

(Assuming by "this fact", you meant monadicity, rather than the explicit description of the monad.)

Mike Shulman (Jun 08 2025 at 17:23):

I expect you could deduce the explicit description of the monad from the fact that a strict factorization system is a distributive law in Prof plus general facts about lifted monads from distributive laws. (Maybe that's what you had in mind originally.) But I don't know a reference.

Amar Hadzihasanovic (Jun 08 2025 at 19:03):

Nathanael Arkor said:

This is an instance of the fact that restriction along [[dominant functors]] (and, in particular, identity-on-object functors) is monadic. E.g. a reference is Example A4.2.7(b) of the Elephant.

The result in the Elephant seems to say that restriction along dominant functors is comonadic, not monadic; indeed, it says it's a sufficient condition for the induced geometric morphism to be a [[surjective geometric morphism]] which means that the adjunction is comonadic; is the nLab wrong then, or is there something else about dominant functors which makes the adjunction both monadic and comonadic?

Amar Hadzihasanovic (Jun 08 2025 at 19:22):

Ah, of course, the restriction functor is both a left and a right adjoint and it reflects isomorphisms, so the crude monadicity theorem applies. (Not spelled out in the Elephant but I guess one is expected to connect the dots.)

Nathanael Arkor (Jun 08 2025 at 21:34):

I think the issue is really with the nLab page, which is leaving the missing assumption of the crude monadicity theorem implicit; I've updated it.