nLab essentially surjective functor

Essentially surjective functors

Essentially surjective functors


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”.


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.

In homotopy type theory

A functor F:CDF : 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.



  • 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.

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Last revised on May 1, 2023 at 11:04:04. See the history of this page for a list of all contributions to it.