nLab
essentially surjective functor

Essentially surjective functors

Idea

A functor F:CDF\colon C \to D is essentially surjective, or essentially surjective on objects (sometimes abbreviated to eso), if it is surjective on objects “up to isomorphism”.

Definition

F:CDF\colon C \to D is essentially surjective if for every object yy of DD, there exists an object xx of CC and an isomorphism F(x)yF(x) \cong y in DD.

Examples

Properties

  • Strengthening the last example, there is an orthogonal factorization system (in the up-to-isomorphism strict sense) on CatCat, in which eso functors are the left class and fully faithful functors are the right class.

    This is an “up-to-isomorphism” version of the bo-ff factorization system, which is a 1-categorical orthogonal factorization system on CatCat in which the left class consists of bijective-on-objects functors. Thus essentially surjective is a non-evil version of “bijective on objects”, i.e. the version which views CatCat as a bicategory.

    In particular, while a functor factors uniquely-up-to-isomorphism as a b.o. functor followed by a fully faithful one, it factors only uniquely-up-to-equivalence as an e.s.o. functor followed by a fully faithful one. Since b.o. functors are also e.s.o., any (eso,ff) factorization of some functor is equivalent to its (bo,ff) factorization.

  • In any 2-category there is a notion of eso morphism which generalizes the essentially surjective functors in Cat. In a regular 2-category, these form a factorization system in a 2-category together with the ff morphisms.

Revised on May 11, 2012 12:05:51 by Urs Schreiber (82.169.65.155)