Descent of singular cohomology - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:24:23Z http://mathoverflow.net/feeds/question/30500 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30500/descent-of-singular-cohomology Descent of singular cohomology fs1504 2010-07-04T11:03:15Z 2010-07-04T11:03:15Z <p>When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the author of the proof I have read observes that the sheafifications $\mathcal{C}^n$ of the singular cochain complexes <code>$C^n(-)=\hom_{Ab}(\mathbb{Z}\hom_{Top}(\Delta^n,-),\mathbb{Z})$</code> form an injective resolution $$0\to\mathbb{Z_X}\to\mathcal{C}^0\to\mathcal{C}^1\to\ldots$$ of $\mathbb{Z}_X$.</p> <p>Why must one sheafify the singular cochain complexes? Aren't they sheaves since they satisfy descent (= have the excision property) and "sheaf=presheaf+descent"?</p>