A remark on singular cohomology and sheaf cohomology. (English) Zbl 1506.18005
A topological space \(X\) is said to be cohomologically locally connected with respect to an abelian group \(A\) if for all \(x\in X\) and all \(k\in\mathbb{Z}\), we have
\[
\varinjlim_{x\in U}\boldsymbol{H}_{\mathrm{sing}}^{k}(U,x;A)=0
\]
This paper aims to establish the following theorem.
Theorem. Suppose that a topological space \(X\) is cohomologically locally connected with respect to an abelian group \(A\). Then sheaf cohomology and singular cohomology of \(X\) with coefficients in \(A\) are canonically isomorphic.
Theorem. Suppose that a topological space \(X\) is cohomologically locally connected with respect to an abelian group \(A\). Then sheaf cohomology and singular cohomology of \(X\) with coefficients in \(A\) are canonically isomorphic.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18B25 | Topoi |
| 55N30 | Sheaf cohomology in algebraic topology |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
| 18G35 | Chain complexes (category-theoretic aspects), dg categories |
| 55N10 | Singular homology and cohomology theory |
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