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A good answer to both questions is provided by the following variant of the Gelfand duality for commutative von Neumann algebras, which shows that the following categories are equivalent: The categor …

We can give a complete classification of (candidates for) delta-distributions on a smooth manifold $M$ at point $p$. Specifically, a delta-distribution is a smooth linear functional on the space of sm …

The original two papers by Leray from 1946 and 1950: Jean Leray, L’anneau d’homologie d’une représentation. Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946), 1366–1368. …

A smooth section $X$ of a vector bundle (e.g., a smooth vector field) over an open subset $U⊂M$ can be presented in the form $X=g Y$, where $g$ is a smooth function on $U$ and $Y$ is a smooth section …

I have not yet found the statement of your answer by localization and construction of the sheaf, in your own publications and available texts, or elsewhere. To the best of my knowledge, there is not …

Such a notion of a principal $\def\bC{{\bf C}}\bC$-bundle (when $\bC$ is a topological or simplicial category, or a Segal space) is available in Definition 6.1 of the paper Classifying spaces of infin …

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 …

To summarize the discussion in the comments: There are two nontrivial Grothendieck topologies used in the definition of a concrete sheaf: the Grothendieck topology T used to define sheaves, and the G …

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 …

This answer provides a positive answer to a refinement of the original question. Recall that two closed differential $k$-forms $ω_0$, $ω_1$ on a smooth manifold $M$ have the same de Rham cohomology cl …

That is, what algebraic structure captures this kind of groupoid-with-restriction and how do we describe its action on a given sheaf more precisely? This structure is well known and has many equival …

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By a “base” in this answer I mean what appears to be the most common definition: a c …

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 …

Both cases ($F$ is a sheaf of abelian topological groups or abelian Lie groups) can be treated using the same machinery. The Yoneda embedding embeds abelian Lie groups as a fully faithful subcategory …

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 $$\mathop{\rm colim} F(U) \to F(K)$$ …