nLab surjective on objects functor

Idea

A functor $F:C\to D$ is surjective on objects if for each object $y\in D$ there is an object $x\in C$ such that $F(x)=y$.

This is of course not an invariant notion; the corresponding invariant notion is essentially surjective functor. However, surjective-on-objects functors are sometimes useful when doing strict 2-category theory.

Properties

Surjective-on-objects functors are the left class in an orthogonal factorization system on Cat whose right class consists of the injective-on-objects and fully faithful functors.

Related pages

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

References

• Ross Street, Two-dimensional sheaf theory refers to surjective-on-objects functors as acute and their corresponding right class as chronic.