nLab essentially surjective (infinity,1)-functor

Definition

An $(\infty,1)$-functor $F : C \to D$ is essentially surjective if the induced functor of the core infinity-groupoids

$$core(F_0) : core(C_0) \to core(D_0)$$

is an effective epimorphism.

An $(\infty,1)$-functor $F : C \to D$ is essentially surjective if the induced functor of the homotopy categories of the $(\infty,1)$-categories

$$h F_0 : h C_0 \to h D_0$$

is essentially surjective

Properties

An (∞,1)-functor which is both essentially surjective as well as full and faithful (∞,1)-functor is precisely an equivalence of (∞,1)-categories.

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