nLab equality and equivalence - contents

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.

  • equivalence of categories, adjoint equivalence, weak equivalence of internal categories, Morita equivalence, equivalence of 2-categories, equivalence of (∞,1)-categories

principle of equivalence

• univalence

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

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