Definition of "presheaf"

presheaf

noun
plural presheaves
1. (category theory, algebraic geometry) A contravariant functor whose domain is a category whose objects are open sets of a topological space and whose morphisms are inclusion mappings. The functorial images of the open sets are sets of things called sections which are said to be "over" those open sets. The (contravariant) functorial images of those inclusion mappings are functions which are called restrictions.
  Quotations
  Let X be a topological space. (Following tradition, I will switch from my previous convention of using X to denote an object of a topos.) Write Open(X) for its poset of open subsets. A presheaf on X is a functor F : O p e n ( X ) op → S e t {\displaystyle F:\mathbf {Open} (X)^{\mbox{op}}\rightarrow \mathbf {Set} } . It assigns to each open subset U a set F(U), whose elements are called sections over U (for reasons to be explained). It also assigns to each open V ⊆ U {\displaystyle V\subseteq U} a function F ( U ) → F ( V ) {\displaystyle F(U)\rightarrow F(V)} , called restriction from U to V and denoted by s → s | V {\displaystyle s\rightarrow s|_{V}} . I will write Psh(X) for the category of presheaves on X.Examples 3.1 i. Let F(U) = {continuous functions U → R {\displaystyle U\rightarrow \mathbb {R} } }; restriction is restriction.
  2011 June 27, Tom Leinster, “An informal introduction to topos theory”, in arXiv.org‎, Cornell University Library, retrieved 2018-03-18