presheaf

English

Etymology

pre- +‎ sheaf

Noun

presheaf (plural presheaves)

1. (category theory, sheaf theory) An 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 ${\mathcal {F}}$ 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 ${\mathcal {F}}$ 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 $A\to B$ under ${\mathcal {F}}$ is a morphism ${\mathcal {F}}(B)\to {\mathcal {F}}(A)$, called the restriction from $B$ to $A$ and denoted $\operatorname {res} _{B,A}$ or $|_{B,A}$.
   • 2011 June 27, Tom Leinster, “An informal introduction to topos theory”, in arXiv.org‎^[1], Cornell University Library, retrieved 2018-03-18:

     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 $F:\mathbf {Open} (X)^{\mbox{op}}\rightarrow \mathbf {Set}$. 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 $V\subseteq U$ a function $F(U)\rightarrow F(V)$, called restriction from U to V and denoted by $s\rightarrow s|_{V}$. I will write Psh(X) for the category of presheaves on X.
     
     Examples 3.1      i. Let F(U) = {continuous functions $U\rightarrow \mathbb {R}$}; restriction is restriction.

Usage notes

• If the base space is denoted as X and the presheaf's codomain is denoted A, then the presheaf is said to be "on X, with values in A".

Hyponyms

• sheaf