Definition N23.1 (Functor). A functor consists of a map
which takes every and assigns an object (think sets groups), and, for every , a map on morphisms
i.e. every is mapped to an (think like: set unions cartesian group products), which all must respect
Definition N23.2 (Contravariant functor). A contravariant functor is one such that each (for all ) maps to
which reverses the morphism. Therefore, for all ,.
Example N23.3 (Duality). For a vector space over ,, whose elements are linear maps . This gives
such that a linear map maps to , with .
“Decorated sets” rings, groups, vector spaces, topological spaces, etc.
Forgetful functors
Given some decorated sets , we can do
(thereby losing structure and operations).