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Is sheaf cohomology an invariant of the weak homotopy type? More precisely let $R$ be a commutative ring and $f:X\rightarrow Y$ a weak homotopy equivalence. Does it follow, that the induced maps $H^n(Y,\underline R) \rightarrow H^n(X,\underline R)$ are isomorphisms?

If $f$

Edit: Since any space is an actual weakly homotopy equivalence or spaces equivalent to a CW-complex, CW-complexes are locally contractible and for locally contractible spaces sheaf and singular cohomology coincide, a positive answer to this is truequestion would imply that sheaf cohomology and singular cohomology coincide for any space. This seems unlikely, but I don't know a counter example.

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Is sheaf cohomology an invariant of the weak homotopy type? More precisely let $R$ be a commutative ring and $f:X\rightarrow Y$ a weak homotopy equivalence. Does it follow, that the induced maps $H^n(Y,\underline R) \rightarrow H^n(X,\underline R)$ are isomorphisms?

If $f$ is an actual homotopy equivalence or spaces are locally contractible this is true.

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# Sheaf cohomology invariant of weak homotopy type?

Is sheaf cohomology an invariant of the weak homotopy type? More precisely let $R$ be a ring and $f:X\rightarrow Y$ a weak homotopy equivalence. Does it follow, that the induced maps $H^n(Y,\underline R) \rightarrow H^n(X,\underline R)$ are isomorphisms?

If $f$ is an actual homotopy equivalence or spaces are locally contractible this is true.