Obtaining a morphism of sheaves from homotopy data

categorist · science forum beginner Joined: 17 Jul 2006 Posts: 1

Let X be a topological space, and let F and G be sheaves of spaces on
X. Suppose that

h:F(X)-->G(X)

is a map of global sections with the following property. For any open
subset U in X, there is a map

k:F(U)-->G(U)

such that k is homotopic to the restriction of h. That is, both h and
k restrict to vertices in the mapping space Map(F(X),G(U)), and there
is a path between them. Assume moreover that the space of such k's is
contractible for every U.

Can you find a map of sheaves h':F-->G such that h'(X) is homotopic to
h in Map(F(X),G(X))?

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