nLab
equivalence of infinitygroupoids
Context
Homotopy theory
homotopy theory
Introductions
Background
Variations
Definitions
Paths and cylinders
Homotopy groups
Theorems
Equality and Equivalence
equivalence

equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)

identity type, equivalence in homotopy type theory

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

EulerLagrange equation, Einstein equation, wave equation

Schrödinger equation, KnizhnikZamolodchikov equation, MaurerCartan equation, quantum master equation, EulerArnold equation, Fuchsian equation, FokkerPlanck equation, Lax equation
An equivalence in an (∞,1)category in ∞Grpd, hence ann equivalence of (∞,1)categories between (∞,1)categories that happen to be ∞groupoids.
In terms of presentation by topological spaces or simplicial sets (see at homotopy hypothesis) this is a weak homotopy equivalence.
In terms of presentation by CWcomplexes or Kan complexes, it is a homotopy equivalence.
Revised on January 8, 2016 05:59:44
by
Urs Schreiber
(89.204.130.2)