A subfunctor is a subobject in a functor category.
A subfunctor of a functor between categories and is a pair where is a functor and is a natural transformation such that its components are monic.
In fact one often by a subfunctor means just an equivalence class of such monic natural transformations; compare subobject.
A subfunctor is also called a subpresheaf . A subfunctor of a representable functor is precisely a sieve over the representing object .
In a concrete category with images one can choose a representative of a subfunctor where the components of are genuine inclusions of the underlying sets; then a subfunctor is just a natural transformation whose components are inclusions. The naturality in terms of concrete inclusions just says that for all , . If the set-theoretic circumstances allow consideration of a category of functors, then a subfunctor is a subobject in such a category.
A subfunctor of the identity in a category with images is an often used case: it amounts to a natural assignment of a subobject to each object in . For concrete categories with images then .
Last revised on March 29, 2011 at 23:35:27. See the history of this page for a list of all contributions to it.