essentially surjective functor in nLab

Essentially surjective functors

Essentially surjective functors

Idea

A functor F:C→DF\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:C→DF\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.

In homotopy type theory

A functor F:C→DF : C \to D is essentially surjective if for all y:Dy:D there merely exists an x:Cx:C such that F(x)≅yF(x) \cong y.

Examples

• A functor between discrete categories (or, more generally, skeletal categories) is essentially surjective iff it is a surjective function between the classes of objects.

• Any bijective-on-objects functor or surjective-on-objects functor is essentially surjective.

• A composition of any two essentially surjective functors is essentially surjective.

• If gfg f is essentially surjective, then gg is essentially surjective.

• An essentially surjective functor is additionally fully faithful precisely when it is an equivalence of categories.

• The inclusion functor of a subcategory is essentially surjective precisely when the subcategory is essentially wide.

• Every split essentially surjective functor is essentially surjective. The converse is true for strict functors in the presence of the axiom of choice.

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 the notion of essential surjectivity is a version of “bijective on objects” which does respect the principle of equivalence, i.e. the version which views Cat 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.