Morphism

A morphism is an arrow between objects in a category, subject to composition and identity laws. It generalizes structure-preserving maps across mathematical settings.

Category theory: A mathematical framework built from objects, morphisms, identities, and composition. Morphism is one of the two basic kinds of data in this framework.
Associativity: A law stating that regrouping a sequence of compatible operations does not change its result. Associativity requires either grouping of composed morphisms to yield the same arrow.
Group homomorphism: A function between groups that preserves the group operation. Groups and their homomorphisms form a standard category.
Set-theoretic function: A relation assigning exactly one output to each element of a domain. A function alone need not preserve the structures required of a category's morphisms.
Category: A collection of objects and morphisms with identities and associative composition. A category specifies which morphisms exist and how they compose.
Commutative diagram: A diagram of objects and morphisms in which all paths with the same endpoints compose equally. Commutativity expresses equations between routes built from morphisms.
Continuous function: A function between topological spaces whose inverse images of open sets are open. Continuous functions are the morphisms in the category of topological spaces.
Relation (mathematics): A subset of a Cartesian product that specifies which elements are related. Relations need not be single-valued or obey a category's composition laws.
Domain (category theory): The source object of a morphism. The domain identifies where this arrow starts.
Monomorphism: A morphism that can be canceled on the left in every composable equality. It captures an abstract form of injectivity using composition.

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