Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
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 $f : C \to D$ 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.
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.
equivalence of (∞,1)-categories, adjoint (∞,1)-functor
basic properties of…
Last revised on May 30, 2023 at 08:43:11. See the history of this page for a list of all contributions to it.