morfismid

Morfismid (morphisms) in mathematics are arrows that relate objects within a category. A morfism f: A → B has source A and target B. If f: A → B and g: B → C, the composite g ∘ f: A → C is a morfism. Every object A has an identity morfism id_A: A → A that acts as a neutral element for composition. These rules define a category.

Special morfismid include monomorphisms (left-cancellable), epimorphisms (right-cancellable), and isomorphisms (invertible morphisms). An isomorphism f: A → B has an inverse g: B → A with g ∘ f = id_A and f ∘ g = id_B.

Examples: in groups, morfismid are group homomorphisms; in topology, continuous maps; in linear algebra, linear maps; in ring theory, ring homomorphisms. In each context, composition of structure-preserving maps remains structure-preserving.

In category theory, morfismid are abstract arrows; a functor maps objects to objects and morfismid to morfismid, preserving composition and identities. Natural transformations relate functors by morphisms between their images.

History: the idea of morfismid and categories emerged in the 1940s from the work of Eilenberg and Mac Lane, providing a unifying language for algebra, topology, and beyond.

Notation: morfismid are denoted by letters such as f, g, h with domain and codomain; composition is written g ∘ f; identity morfisms are id_A; categories satisfy associativity and identity laws.