12:04 PM
Does a morphism of schemes $f:X\to Y$ induce a group homomorphism on sheaf cohomology? Like $H^i(X,\mathcal{F})\to H^i(Y,f_*\mathcal{F})$ if $\mathcal{F}$ is a sheaf of groups on $X$, or like $H^i(Y,\mathcal{G})\to H^i(X,f^*\mathcal{G})$ if $\mathcal{G}$ is a sheaf of groups on $Y$? If so, is there a way to reframe this, such that $H^i(-,\mathcal{F})$ is a functor on $S$-schemes (fixing the issue where there is really a different sheaf for each $S$-scheme)?
12:18 PM
@TedE Yes, essentially because $H^i(Y,f^*\mathcal{G})$ is also the cohomology of $Rf_\ast f^*\mathcal{G}$ and the map you want comes from the unit of the (derived) adjunction $\mathcal{G}\to Rf_\ast f^\ast\mathcal{G}$. So for every sheaf $\mathcal{G}$ over $S$ there is a functor sending $f$ to $R f_\ast f^\ast \mathcal{G}$ and then you can take cohomology (I'm skipping a bit of categorical details but hopefully the idea is clear)
2 hours later…
2:26 PM
@SaalHardali I once wrote it down in detail that if you have a nice model category C and a G a set of compact objects, then the underlying $\infty$-category of the Bousfield model structure on $sC$ (with respect to the choice of generators given by G) is that of product-preserving presheaves on G.
I didn't do anything with it in the end, because it didn't seem of much interest (it turns out that in practice one can work with product-preserving presheaves directly and never involve resolution model structures). I can send it to you though.
I think it would be nice if someone worked out an account internal to $\infty$-categories (ie. how to present $P_{\Sigma}(C^{\omega})$ as a localization of $sC$, for say $C$ stable, compactly-generated). I always thought that Lurie's appendix in SAG on Kan complexes in $\infty$-topoi would be somehow relevant.
(Also, to answer your question: for say $C$, stable, one expects that $sC \rightarrow P_{\Sigma}(C^{\omega})$ is a localization along those maps $X_{\bullet} \rightarrow Y_{\bullet}$ such that $[C, X_{\bullet}] \rightarrow [C, Y_{\bullet}]$ is a weak equivalence of simplicial sets for any $C \in C^{\omega}$.)
3:18 PM
Just to confirm:
Given an $S$-scheme, $f:Y\to S$, such that the structure morphism $f$ is affine, and a sheaf $\mathcal{G}$ on $S$, by making use of the isomorphism $H^i(Y,f^*\mathcal{G})\cong H^i(S,f_*f^*\mathcal{G})$, we can get a well defined homomorphism $H^i(S,\mathcal{G})\to H^i(Y,f^*\mathcal{G})$ by making use of the unit of adjunction applied to $\mathcal{G}$, i.e. we have a morphism of sheaves
$$\mathcal{G}\to f_*f^*\mathcal{G},$$
which induces a morphism:
$$H^i(S,\mathcal{G})\to H^i(S,f_*f^*\mathcal{G})\cong H^i(Y,f^*\mathcal{G})$$
Given an $S$-scheme, $f:Y\to S$, such that the structure morphism $f$ is affine, and a sheaf $\mathcal{G}$ on $S$, by making use of the isomorphism $H^i(Y,f^*\mathcal{G})\cong H^i(S,f_*f^*\mathcal{G})$, we can get a well defined homomorphism $H^i(S,\mathcal{G})\to H^i(Y,f^*\mathcal{G})$ by making use of the unit of adjunction applied to $\mathcal{G}$, i.e. we have a morphism of sheaves
$$\mathcal{G}\to f_*f^*\mathcal{G},$$
which induces a morphism:
$$H^i(S,\mathcal{G})\to H^i(S,f_*f^*\mathcal{G})\cong H^i(Y,f^*\mathcal{G})$$
3:29 PM
@PiotrPstrągowski Thanks! I found the statement you stated in Mazel-Gee's thesis which had many other interesting statements in that direction but I wasn't able to find a statement of the form I outlined. The real reason I'm interested in this is I wonder if something analogous can be done for cosimplicial objects in a nice $\infty$-category. So that we can speak about the category of resolutions by totalizations rather than geometric realizations.
4:07 PM
@SaalHardali At a formal level, you can always adjoin a class of limits or a class of colimits to an $\infty$-category. Product-preserving presheaves adjoins all sifted colimits, but one can also adjoin only geometric realizations (and sometimes this is preferable, as in Brantner's thesis).
Dualizing, you can adjoin all totalizations, although objects such that mapping into them takes totalizations to geometric realizations are probably more rare then the dual variant (compact projective). One notable examples is that if $A$ is Grothendieck abelian, then the coconnective derived $\infty$-category $D_{\leq 0}(A)$ is obtained from the subcategory of injectives in $A$ by freely adjoining totalizations (this is in SAG, "Injective objects in stable $\infty$-categories).
4:47 PM
@SaalHardali One more article I can point you towards is Biedermann's "Interpolation categories for homology theories", where he uses resolution model structures on cosimplicial objects to construct a Goerss-Hopkins-type tower for an essentially arbitrary homology theory. This must be somehow dual to what Goerss and Hopkins do, but I've never seen it worked out.
@PiotrPstrągowski Maybe it will be useful to mention that the reason that dualizing this interests me is the possibility of defining non-abelian derived category for coalgebraic structures. For example the category of conilpotent cocommutative coalgebras is in some sense what you get from considering the subcategory of conilpotent cofree cocommutative algebras, freely adjoining totalizations, and then after that freely adjoining filtered colimits.
5:43 PM
@TedE Yes, you can remove all assumptions (no assumptions whatsoever are needed on $Y$, just that $f$ is quasi-compact and quasi-separated), you just need to use $Rf_\ast$ (the derived right adjoint) rather than $f_\ast$.
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