Morphism

A morphism is a map between two mathematical structures of the same kind that preserves the defining operations or relations of that structure.

The precise definition depends on context:

• Between groups : a group homomorphism preserves the group operation.
• Between rings : a ring homomorphism preserves addition and multiplication.
• Between vector spaces : a linear map preserves addition and scalar multiplication.
• Between topological spaces: a continuous map preserves the property of being open (via preimages).
• Between smooth manifolds: a smooth map preserves differentiable structure.

This pattern—objects plus structure-preserving maps—is formalized in category theory , where morphisms are the arrows between objects satisfying composition and identity axioms.

An isomorphism is a morphism with a two-sided inverse that is also a morphism; isomorphic structures are “the same” from the perspective of their defining properties.