nLab equivalence of (infinity,1)-categories

Definition

An (∞,1)-functor between (∞,1)-categories is an equivalence in (∞,1)Cat precisely if it is an essentially surjective (∞,1)-functor and a full and faithful (∞,1)-functor.

When (∞,1)-categories are presented by quasi-categories, an equivalence between them is presented by a weak equivalence in the model structure for quasi-categories.

Properties

Lemma

An (∞,1)-functor $f : C \to D$ is an equivalence in (∞,1)Cat if the following equivalent conditions hold

• On the underlying simplicial sets it is a weak equivalence in Joyal’s model structure for quasi-categories (a “weak categorical equivalence”).

• For every simplicial set $K$ the induced morphism on mapping complexes $f_* \colon SSet(K,C) \to SSet(K,D)$ is a weak categorical equivalence.

• For every simplicial set $K$ the induced morphism $f_* \colon Core(SSet(K,C)) \to Core(SSet(K,D))$ of cores (the maximal Kan complexes) is a weak equivalence in the Kan-Quillen model structure, hence a simplicial weak homotopy equivalence.

• The induced morphism $f_* : Core(SSet(\Delta^1, C)) \to Core(SSet(\Delta^1, D))$ is an equivalence of Kan complexes.

Proof

The equivalence of the first three points is HTT, lemma 3.1.3.2.

Cisinki, theorem 3.9.2, shows the third point can be weakened to taking just $K = \Delta^0, \Delta^1$, so to show equivalence with the fourth point, it suffices to show the $K = \Delta^1$ case implies the $K = \Delta^0$ case.

Suppose that $f_* : Core(SSet(\Delta^1, C)) \to Core(SSet(\Delta^1, D))$ is an equivalence. For each object $d \in D_0$, there is an edge $\varphi : c \to c'$ in $C_1$ such that $f_*(\varphi) \simeq id_d$. This implies $f(c) \simeq d$ and thus $f_*(id_c) \simeq id_d$, and that $\varphi \simeq id_c$.

Thus, $f_*$ restricts to an equivalence between the subcomplexes consisting of the connected components of the identity morphisms. But for any quasi-category $X$, this subcomplex is $\Core(X)^{\Delta^1} \subseteq \Core(X^{\Delta^1})$, and the degeneracy $\Core(X) \to \Core(X)^{\Delta^1}$ is an equivalence. Thus, the induced map $Core(C) \to Core(D)$ is an equivalence.

Related concepts

• equality

• isomorphism

• equivalence

• weak equivalence

• homotopy equivalence, weak homotopy equivalence

• homotopy equivalence of toposes

• equivalence in an (∞,1)-category

• equivalence of categories, adjoint functor

• equivalence of 2-categories, 2-adjunction

• equivalence of (2,1)-categories

• equivalence of (∞,1)-categories, adjoint (∞,1)-functor

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

• Denis-Charles Cisinski, “Higher Categories and Homotopical Algebra”