A morphism is a formal arrow between two objects of a category, the basic unit of category-theoretic structure description. It is defined by a category C = (Ob, Mor, (·)∘(·), (id_A)_(A∈Ob)) in which each morphism f has a domain object A and codomain object B, written f: A → B; morphisms compose (g∘f exists when the codomain of f equals the domain of g), composition is associative, and every object A carries an identity morphism id_A satisfying id_A∘f = f = f∘id_A. Special cases include functions (category Set), group homomorphisms (Grp), and continuous maps (Top); a morphism possessing an inverse is an isomorphism. Parameters: the ambient category (objects, hom-sets, composition law, identities) and the domain/codomain of the arrow. It persists as the foundational vocabulary of category theory as codified by Eilenberg and Mac Lane (1945), through standard textbook treatments (Mac Lane's Categories for the Working Mathematician, Awodey's Category Theory) and its standing use as the common language of structure-preserving maps across algebra, topology, and logic. [formal: probare | substrate: mind | horizon: a life | explicit: yes | epoch: 0.01]
Accepted ontology entry
morphism
A morphism is a formal arrow between two objects of a category, the basic unit of category-theoretic structure description. It is defined by a category C = (Ob, Mor, (·)∘(·), (id_A)_(A∈Ob)) in which each morphism f has a domain object A an…
Definition
Why it is in scope
A human-made formal object of category theory — an arrow between objects in a category — built with composition and identity axioms so it can serve as the universal language of structure-preserving maps. It persists as the foundational vocabulary of category theory across mathematics and computer science.
Names and aliases
- morphismen · CANONICAL
Relations from this entry
- cmrg4vstb00kx2a1nar61eoe7SERVES →
A morphism is a structure-preserving map between objects, a construct built and maintained for the sake of mathematics — it serves mathematical reasoning about structure. Servant (morphism) points at master (mathematics).
Relations to this entry
- cmrnprjk302xpd1nldgu17m3r← INSTANCE_OF
An isomorphism is a morphism with an inverse: in a category, an isomorphism A → B is a morphism f: A→B for which a morphism g: B→A exists with g∘f = id_A and f∘g = id_B. Every isomorphism is a morphism — a category theorist says 'an isomorphism is a (special) morphism', and the isomorphism's own accepted definition ('a bijective mapping ... with a mapping in the reverse direction') describes exactly a morphism possessing an inverse. Law 11e nearest kind: morphism is the closest existing kind (no function/bijection entry stands nearer). Specific → general.
- category← CONTAINS
Category is defined as the formal quadruple (Ob, Mor, composition, identities). The morphism collection Mor is a constituent component of the category structure; the definition of category specifies its parameters include the collection of morphisms between objects. Removing morphisms from the category definition eliminates the structure. Part-of direction: category contains morphism as constituent.
Record identity
- Created
- Sep 4, 2026, 9:59 AM UTC
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- 109090bc57b6004929f82dd9b310659583b6e3f7a7f477153bd5ff5e2cc67518