**Tags**

algebra, arrow, category, function, geometry, identity, inclusion, intersection, map, metacategory, object, open, presheaf, sheaf, subset, topological space, topology

**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 p_{UV}:* F*(U)->

*(*

**F***V*), such that

condition 1) p_{UU} is the identity arrow of * F*(U),

condition 2) p* _{UW}*=p

*p*

_{VW}*where p*

_{UV}*and p*

_{VW}*are defined,*

_{UV}condition 3) for any open covering {*V _{i}*} of any open subset

*U*of

*X*, and for any element

*s*of

*(*

**F***U*) such that p

*(*

_{UVi}*s*)=0 for all

*i*,

*s*=0 and

condition 4) for any open covering {*V _{i}*} of any open subset

*U*of

*X*, and for any set {

*s*} of elements of

_{i}*(*

**F***U*) such that p

*int*

_{UVi}*V*(

_{j}*s*)=p

_{i}*int*

_{UVi}*V*(

_{j}*sj*) and

*s*is in

_{i}*(*

**F***V*) for all

_{i}*i*, there exists an element

*s*of

**(**

*F**U*) such that p

*(*

_{UVi}*s*)=

*s*for each

_{i}*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 p_{UV}:* F*(U)->

*(*

**F***V*), such that

condition 1) p_{UU} is the identity arrow of * F*(U) and

condition 2) p* _{UW}*=p

*p*

_{VW}*where p*

_{UV}*and p*

_{VW}*are defined.*

_{UV}