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Patrick Nicodemus (Aug 19 2022 at 09:47):image.png
In Jon Beck's thesis he makes this comment. Theorem 3.1 refers to Artin's "Grothendieck Topologies" and the theorem is that, for presheaves of Abelian groups the higher Cech cohomology functors are the right derived functors of .
Patrick Nicodemus (Aug 19 2022 at 09:47):Does anyone have a good explanation of what he is referring to with the close relationship between monads and Grothendieck topologies
Morgan Rogers (he/him) (Aug 19 2022 at 09:52):The sentence is too vague to be sure: does he mean that there is a close relationship between monads on a category and Grothendieck topologies on the same category (it's unclear to me what relationship he would be referring to in this case)? Is he observing that sheaves are a reflective subcategory of presheaves? Is he observing that sheaves of groups are internal groups and so monadic over the category of sheaves? Difficult to say.
Callan McGill (Aug 20 2022 at 17:39):I believe it is referring to the idea of monadic descent along a covering morphism: https://ncatlab.org/nlab/show/monadic+descent I don’t know an immediate reference that explains this in terms of covers but I suspect it is in Sheaves in Geometry and Logic