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I am interested for many years in using localizations (typically having fully faithful right adjoint) as replacements for open sets in noncommutative algebraic geometry. There are some interesting Čech-like cohomologies lurking there as well (for abelian case known for about 20 years, for nonabelian wait for another paper of mine).
Now I am trying to do, with some ideas from correspondence with colleagues some more new cases, some of which to my surprise are not known. In particular there is lots of issues about descent for functors where both the domain and codomain categories are given by covers by localizations, that is families of localizations such that the induced inverse image functor to the product of the localizations is comonadic. Before tackling that case properly, I am trying to fix some holes in my understanding of the case of endofunctors.
A localization having fully faithful right adjoint is equivalent to a Eilenberg-Moore category over an idempotent monad. For any localization functor, there is a notion of localization compatible endofunctor, which Lunts and Rosenberg introduced in unpublished but available Max Planck Preprint in 1996, in order to study the differential operators on geometries given by “categories of quasicoherent sheaves”. Their notion of compatibility does not involve coherences, and I prefer to work with coherences. I have shown that the coherences actually follow, if one takes for compatibility a distributive law between the idempotent monad $Q$ and the endofunctor $G$, of the form $QG\Rightarrow GQ$. More precise statement is at distributive law for idempotent monad (zoranskoda) (cf. also new entry compatible localization with some background references, some on different notions of compatibilities).
The compatibility which is a property in Lunts-Rosenberg becomes a distributive law in my work, what is in general a structure; but a (seamingly new) theorem says that the coherent version is also just a property, namely the distributive law is unique when it exists (“strict compatibility” Edit: It seems to me that the existence of a distributive law is a bit stronger than Lunts-Rosenberg compatibility.) The law is then also invertible what means that its inverse is the other type distributive law $GQ\Rightarrow QG$. But what I don’t know is if every other type distributive law $GQ\Rightarrow QG$ for an idempotent monad $Q$ is invertible and hence coming from the inverse of a unique distributive law $QG\Rightarrow GQ$ or there are genuine non-invertible counterexamples ? Any ideas ?
This is not an answer to the main question, but unless I’m missing something, the uniqueness of the distributive law QG ⇒ GQ is immediate from the fact that such a distributive law is the same as a lifting of G to the category of Q-algebras, since when Q is idempotent that category is just a full subcategory of the ambient category and hence there can be only one such “lifting” (i.e. when G preserves that subcategory).
Correct Mike, but the same truism, with a different proof holds for distributive laws in other direction, I mean $GQ\Rightarrow QG$ for which one does not have precisely that interpretation.
Can’t a distributive law in the other direction be identified with a lifting of G to the Kleisli category of Q, which is the same as the category of Q-algebras when Q is idempotent?
I don’t think so (unless there is something very special in idempotent case). The other direction would lift $Q$ to $G$-algebras instead (at least when $G$ is a monad, and we have distributive law in that sense).
Oops, I meant a co-lifting, i.e. an endofunctor G’ of the Kleisli category such that G’ F = F G, where F is the left adjoint from the ambient category to the Kleisli category. I think such a thing ought also to be unique when it exists in the idempotent case.
The reason I’m so sure that something like this is true is that the correspondence between distributive laws and liftings works in any 2-category, and Eilenberg-Moore objects in $Cat^{op}$ are the same as Kleisli objects in Cat, while reversing the order of the functors means passing to $Cat^{op}$, and so should correspond to replacing the Eilenberg-Moore category by the Kleisli one.
It could be, I roughly understand your argument and if I look at my argument (which I did not present yet) it is diagrammatic via certain commutative triangles, but they indeed can be dualized. Good, I mean to have independent reasoning to be safer on the uniqueness side. What is left is still the two main questions: are there any non-invertible distributive laws in either direction; and if the incoherent compatibility (that is $Q^* G = G' Q^*$ for some $G' : B\to B$) implies existence of a distributive law of either kind.
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