nLab
equivalence in an (infinity,1)-category
Contents
Context
$(\infty,1)$ -Category theory
(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

Equality and Equivalence
equivalence

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

natural equivalence , natural isomorphism

gauge equivalence

Examples.

principle of equivalence

equation

fiber product , pullback

homotopy pullback

Examples.

linear equation , differential equation , ordinary differential equation , critical locus

Euler-Lagrange equation , Einstein equation , wave equation

Schrödinger equation , Knizhnik-Zamolodchikov equation , Maurer-Cartan equation , quantum master equation , Euler-Arnold equation , Fuchsian equation , Fokker-Planck equation , Lax equation

Contents
Definition
For $C$ a quasi-category , a morphism $f : x \to y$ in $C$ (an edge in the underlying simplicial set ) is an equivalence if its image in the homotopy category $Ho(C)$ is an isomorphism .

Equivalently, $f$ is an equivalence if it is the image of a functor of quasi-categories (i.e. a map of simplicial sets) out of the nerve $N(J)$ , where $J$ is the interval groupoid . This is a quasi-categorical version of the general theorem-schema in higher category theory that any equivalence can be improved to an adjoint equivalence .

Last revised on August 5, 2020 at 16:38:11.
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