unitpreservation

Unit preservation, sometimes written as unit-preservation, is the property of a map, morphism, or functor to preserve the identity element (the unit) of a given algebraic or categorical structure. In many areas, preserving the unit is part of preserving the overall structure.

In algebra, a ring homomorphism φ: R → S that satisfies φ(1_R) = 1_S is called unital or unit-preserving; a homomorphism that does not require this is non-unital. The same idea applies in monoids, where a monoid homomorphism must send the unit element to the unit.

In category theory, for monoidal categories, a functor F is strong monoidal if it comes equipped with natural isomorphisms F(X ⊗ Y) ≅ F(X) ⊗' F(Y) and F(I) ≅ I', where I and I' are the unit objects. A unit-preserving (or strictly unit-preserving) monoidal functor preserves the unit object on the nose, i.e., F(I) = I', whereas a non-strict one preserves it up to an isomorphism.

In other algebraic contexts, a homomorphism is often required to preserve designated constants or units, and failure to do so can lead to loss of structure. For example, the zero map does not preserve the unit in ring contexts.

See also: unit element, unital homomorphism, strong monoidal functor, monoidal category.