morphism

== English == === Etymology === Generalised from isomorphism, etc. === Pronunciation === (Received Pronunciation) IPA(key): /ˈmɔːfɪzəm/ (General American) IPA(key): /ˈmɔɹfɪzəm/ Rhymes: -ɔː(ɹ)fɪzəm === Noun === morphism (plural morphisms) (mathematics, category theory) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises. 1982, Israel Program for Scientific Translations (translator), Lev J. Leifman (editor of translation), N. N. Čencov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Translations of Mathematical Monographs, Volume 53, page 50, 1° The composition of two morphisms is defined if and only if the final object of the first morphism is the initial object of the second. This composition is also a morphism, whose initial object is the initial object of the first morphism and whose final object is the final object of the second. 1992, Terrance Brown (translator), Gil Henriques, Chapter 13: Morphisms and Transformations in the Construction of Invariants, Terrance Brown (translator), Jean Piaget, Gil Henriques, Edgar Ascher (editors), Morphisms and Categories: Comparing and Transforming, page 198, In certain extreme cases in mathematics, the synthesis of morphisms and of transformations is so intimate that one can speak of a veritable fusion. […] Essentially, categories are sets of morphisms organized into operatory systems. (biology) Being or having distinct variants of a plant or animal species in the same locale; polymorphism. ==== Usage notes ==== In mathematics, this word is largely interchangeable with homomorphism. However, one uses morphism almost exclusively in the context of category theory, while homomorphism is more common in algebra. ==== Synonyms ==== (category theory): arrow, map ==== Derived terms ==== ==== Related terms ==== ==== Translations ==== === See also === bijection function functor isometry measurable function