the state of having a specified shape or form

(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.

plural of morphism

(category theory) The terminal object of a comma category from a functor to a fixed object; or, dually, the initial object of a comma category from a fixed object to a functor.

(category theory) A morphism from an object to the product of that object with itself, which morphism is induced by a pair of identity morphisms of the said object.

(category theory) An isomorphism between a pair of products which have the same components but in swapped order, which isomorphism commutes with the two associated product diagrams.

(category theory) A unique morphism corresponding to each object of a category, which has its domain equal to its codomain, and which composed with any morphism (with which it is composable) gives that same morphism.

(category theory) A morphism from an object to the subobject classifier which corresponds to a unique subobject of the said object, which subobject is the pullback, along this morphism, of the "true" global element of the subobject classifier.

plural of twist morphism

plural of identity morphism