presheaf

== English == === Etymology === From pre- +‎ sheaf. === Noun === presheaf (plural presheaves or presheafs) (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 F {\displaystyle {\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 F {\displaystyle {\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 → B {\displaystyle A\to B} under F {\displaystyle {\mathcal {F}}} is a morphism F ( B ) → F ( A ) {\displaystyle {\mathcal {F}}(B)\to {\mathcal {F}}(A)} , called the restriction from B {\displaystyle B} to A {\displaystyle A} and denoted res B , A {\displaystyle \operatorname {res} _{B,A}} or | B , A {\displaystyle |_{B,A}} . ==== 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 ==== Derived terms ==== === References ===