Search Results

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. …

Here is one possible way to proceed. (Caveat lector: I haven't checked all the details carefully.) Denote by S the sphere spectrum and by N the (contractible) spectrum that implements a nullhomotopy …

Smooth manifolds are affine, thus the sheaf of smooth functions is determined by its global sections. Now C^∞(M×N)=C^∞(M)⊗C^∞(N). The tensor product here is the projective tensor product of complete l …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

the constant sheaf Z is NOT a homotopy sheaf (even though it is an ordinary sheaf) This type of phrasing is ambiguous and is probably responsible for the confusion. In this sentence, Z is used to re …

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

An atlas is a sheaf on the site of cartesian spaces (the site with objects R ), such that ... One can certainly define smooth manifolds in such terms. The cartesian site has finite-dimensional …

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 …

There are many such results. Consider some smooth manifolds M and N. The internal hom Hom(M,N) is a sheaf on smooth manifolds. We can compute its tangent bundle, and it turns out that the tangent spac …