+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Equality and Equivalence +--{: .hide} [[!include equality and equivalence - contents]] =-- #### Foundations +-- {: .hide} [[!include foundations - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Disambiguation in [[logic]], [[higher category theory]], and the [[foundations]] of mathematics, _equality_ is a notion of when two objects of a collection are considered to be the same objects. Here is a list of distinctions between different notions of _equality_, in different contexts, where possibly all the following pairs of notions are, in turn, "the same", just expressed in terms of different terminologies: * the difference between [[axiom|axiomatic]] and [[definition|defined]] equality; * the difference between [[equality]] and [[identity]], * the difference between intensional and extensional equality, * the difference between [[propositional equality]], [[judgmental equality]], and [[typal equality]], see also [[equality in type theory]] * the difference between equality and [[isomorphism]], * the difference between equality and [[equivalence]], * the possibility of operations that might not preserve equality. ## Related concepts * **equality**, [[equation]] * [[definition]], [[assignment]], [[equality]] * [[inequality]] * [[denial inequality]] * [[apartness relation]] * [[antithesis interpretation]] * [[isomorphism]] * [[equivalence]] * [[weak equivalence]] * [[homotopy equivalence]], [[weak homotopy equivalence]] * [[equivalence in an (∞,1)-category]] * [[equivalence of (∞,1)-categories]] * [[identification]], [[unique identification]] [[!include logic symbols -- table]] category: disambiguation [[!redirects equal]] [[!redirects equals]] [[!redirects equality]] [[!redirects equalities]]