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

Posted: Mon Jul 17, 2006 8:15 pm Post subject:
Obtaining a morphism of sheaves from homotopy data



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))? 
