nLab
subfunctor

Contents

Definition

A subfunctor is a subobject in a functor category.

A subfunctor of a functor G:CDG:C\to D between categories CC and DD is a pair (F,i)(F,i) where F:CDF:C\to D is a functor and i:FGi:F\to G is a natural transformation such that its components i M:F(M)G(M)i_M:F(M)\to G(M) 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 Hom(,x)Hom(-,x) is precisely a sieve over the representing object xx.

Properties

In a concrete category with images one can choose a representative of a subfunctor where the components of ii 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 f:cdf:c\to d, F(f)=G(f)| F(c)F(f)=G(f)|_{F(c)}. If the set-theoretic circumstances allow consideration of a category of functors, then a subfunctor is a subobject in such a category.

A subfunctor (F,i)(F,i) of the identity id C:CCid_C:C\to C in a category with images is an often used case: it amounts to a natural assignment cF(c)icc\mapsto F(c)\stackrel{i}\hookrightarrow c of a subobject to each object cc in CC. For concrete categories with images then F(f)=f| F(c)F(f)=f|_{F(c)}.

Revised on March 29, 2011 23:35:27 by Toby Bartels (69.171.178.86)