presheaf

plural presheaves · plural presheafs

1. sheaf theorycategory theoryAn abstract mathematical construct which associates data to the open sets of a topological space, generalizing the situation of functions, fiber bundles, manifold structure, etc. on a topological space (but not necessarily in such a way as to make the local and global data compatible, as in a sheaf). Formally, A contravariant functor ℱ whose domain is a category whose objects are open sets of a topological space (called the base space or underlying space) and whose morphisms are inclusion mappings. The image of each open set under ℱ is an object whose elements are called sections, and are which are said to be over the given open set; the image of each inclusion map A→B under ℱ is a morphism ℱ(B)→ℱ(A), called the restriction from B to A and denoted operatorname res_(B,A) or |_(B,A).

   1 more example

   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#58;#92;mathbf#123;Open#125;(X)#92;mbox#123;op#125;#92;rightarrow#92;mathbf#123;Set#125;. 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#92;subseteqU a function F(U)#92;rightarrowF(V), called restriction from U to V and denoted by s#92;rightarrows#124;#95;V. I will write Psh(X) for the category of presheaves on X. Examples 3.1 i. Let F(U) = {continuous functions U#92;rightarrow#92;mathbb#123;R#125;}; restriction is restriction.

   2011 June 27, Tom Leinster, “An informal introduction to topos theory”, in arXiv.org, Cornell University Library, retrieved 18 Mar 2018