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The original reference for such results is Proposition 3.3.3 on page 120 in Fabien Morel, Vladimir Voevodsky, A^1-homotopy theory of schemes, Publications mathématiques de l’I.H.É.S., tome 90 (1999), …

Is there a construction of the spectrum of a ring, where it is defined as a locale or Grothendieck site generated by the D(f), and where the relation that D(fk)=D(f) is definitional? Yes, the Zarisk …

The manifold $\def\Hom{\mathop{\rm Hom}}\def\R{{\bf R}}\Hom(\R^{0|2},M)$ is isomorphic to the pullback of the parity-reversed bundle vector bundle $TM⊕TM$ along the projection map $TM→M$. This is Lemm …

Is there a Cech-like way of describing the (hyper)cohomology H∙(X,M∙) or, even better, the complex Rf∗M∙ for some map f? Yes, the Verdier hypercovering theorem allows one to compute sheaf cohomology …

As indicated in the comments, the notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\HH^p(X,F)$ is defined (for example) by Milne in Étale Cohomology as the $p$th right derived functor of the inclusi …

Roughly speaking, a topos in the sense of Grothendieck is the category of sheaves on a kind of generalised space whose “points” may have non-trivial automorphisms. Question 1: What does that mean? …

Question: does the vector bundle generated by this sheaf on X∖Sing is isomorphic to the cotangent bundle on X∖Sing is ? If not what is the sheaf we should define on X to get the cotangent bundle on X …

One can define an analogue of the crystalline topos for smooth manifolds. This is known as the de Rham stack of $M$. One of the easiest constructions of the de Rham stack embeds smooth manifolds fully …

Yes, both Theorem A and Theorem B are special cases of a more general construction. Denote by $R$ the category of commutative unital C*-algebras or the category of commutative rings. Denote by $R'$ th …

By Theorem II.9.5 in Bredon's Sheaf Theory, for any closed subset $K$ of a paracompact space $X$ and for any sheaf of abelian groups $F$ on $X$, the canonical map $$\mathop{\rm colim} F(U) \to F(K)$$ …

Since Čech cohomology is mentioned, I presume that $C$ is the category of open subsets of a topological space. More generally, we can assume $C$ to be an arbitrary site. In this case, the answer to bo …

Given any von Neumann algebra $M$, we can define its noncommutative $\def\L{{\cal L}} \L^p$-spaces $\L^p(M)$ for any $\def\C{{\bf C}} p∈\C$ such that $\Re p≥0$. Here I use the notation $\L^p:={\rm L}^ …

For sufficiently nice topological spaces $X$ (e.g., locally connected for the last two categories to be equivalent, and semilocally simply connected and locally path-connected for all three to be equi …

Virtually any kind of algebraic structure (e.g., group, ring, module, vector space, affine space, etc.) leads to a stack in categories whose objects are bundles of such structures and morphisms are fi …

Is the functor H^n_dR:Man→Set a sheaf with respect to open cover topology on Man? As already pointed out in the comments, the answer is no for n>0, yes for n=0. "in what way is cohomology a sheaf" …