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Sheaves

Let C be a category with objects ob(C), let X be a topological space, and define Top(X) to be the category whose objects are the open subsets of (X) and whose arrows are the inclusion maps between open subsets of (X). A sheaf on X of objects of C consists of

i) for every open subset U of X, an object F(U) of C, and

ii) for every inclusion of an open subset V of X in an open subset U of X, an arrow pUV:F(U)->F(V), such that

condition 1) pUU is the identity arrow of F(U),

condition 2) pUW=pVWpUV where pVW and pUV are defined,

condition 3) for any open covering {Vi} of any open subset U of X, and for any element s of F(U) such that pUVi(s)=0 for all i, s=0 and

condition 4) for any open covering {Vi} of any open subset U of X, and for any set {si} of elements of F(U) such that pUViintVj(si)=pUViintVj(sj) and si is in F(Vi) for all i, there exists an element s of F(U) such that pUVi(s)=si for each i.

Presheaves

The definition of a presheaf is weaker than that of a sheaf.  A preasheaf on X consists of

i) for every open subset U of X, an object F(U) of C, and

ii) for every inclusion of an open subset V of X in an open subset U of X, an arrow pUV:F(U)->F(V), such that

condition 1) pUU is the identity arrow of F(U) and

condition 2) pUW=pVWpUV where pVW and pUV are defined.

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