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equivalence of (infinity,1)-categories

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Context

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

Equality and Equivalence

Contents

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

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=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…

References

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