nLab essentially surjective functor

Idea

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

Definition

$F\colon C \to D$ is essentially surjective if for every object $y$ of $D$, there exists an object $x$ of $C$ and an isomorphism $F(x) \cong y$ in $D$.

In homotopy type theory

A functor $F : C \to D$ is essentially surjective if for all $y:D$ there merely exists an $x:C$ such that $F(x) \cong y$.

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 $g f$ is essentially surjective, then $g$ 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 $Cat$, 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 $Cat$ 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.

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor