equivalence of (infinity,1)-categories



(,1)(\infty,1)-Category theory

Equality and Equivalence



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.



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


The equivalence of the first three points is HTT, lemma

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

Suppose that f *:Core(SSet(Δ 1,C))Core(SSet(Δ 1,D))f_* : Core(SSet(\Delta^1, C)) \to Core(SSet(\Delta^1, D)) is an equivalence. For each object dD 0d \in D_0, there is an edge φ:cc\varphi : c \to c' in C 1C_1 such that f *(φ)id df_*(\varphi) \simeq id_d. This implies f(c)df(c) \simeq d and thus f *(id c)id df_*(id_c) \simeq id_d, and that φid c\varphi \simeq id_c.

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

basic properties of…


Last revised on May 27, 2020 at 12:38:55. See the history of this page for a list of all contributions to it.