Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
identity type, equivalence of types, definitional isomorphism
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 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 the induced morphism on mapping complexes is a weak categorical equivalence.
For every simplicial set the induced morphism 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 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 , so to show equivalence with the fourth point, it suffices to show the case implies the case.
Suppose that is an equivalence. For each object , there is an edge in such that . This implies and thus , and that .
Thus, restricts to an equivalence between the subcomplexes consisting of the connected components of the identity morphisms. But for any quasi-category , this subcomplex is , and the degeneracy is an equivalence. Thus, the induced map 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.