Objectvalued

Object-valued is a term used in category theory and related areas to describe mathematical constructions that take values in objects of a fixed category rather than in elements of a set. An object-valued construction typically assigns to each input an object of a chosen category C and to each morphism in the input category a corresponding morphism in C, forming a functor into C. This contrasts with set-valued constructions, which assign sets and functions between sets.

A common instance is a C-valued presheaf or sheaf: a contravariant functor from a site into a category C. For example, a sheaf of modules on a topological space can be viewed as a C-valued sheaf where C is the category of modules over a fixed ring. In enriched category theory, one often works with hom-objects in a monoidal category and considers constructions where the values are objects of that enriching category, rather than mere sets.

Object-valued concepts are particularly useful for capturing structured data and algebraic or geometric information in a way that respects morphisms and functoriality. They generalize familiar set-valued notions and enable the study of objects with additional structure, such as modules, algebras, or sheaves, in a uniform categorical framework.

Related notions include presheaves and sheaves (as C-valued rather than Set-valued versions), functors between categories, and broader ideas from topos theory and enriched category theory. See also presheaf, sheaf, functor, category, topos, and enriched category theory for broader context.