Dcategory

Dcategory is a term used in category theory to describe a structure that combines the idea of enrichment or internalization with a fixed base category D. In the literature, the phrase is used in at least two related ways, and the precise meaning can vary by author. The two common interpretations are: an internal category in D, and a category enriched over D.

An internal category in D, often called a D-category in this sense, consists of an object Ob in D representing the objects of the category and an object Mor in D representing the arrows. There are structure maps s, t: Mor → Ob (source and target), a map i: Ob → Mor for identities, and a composition map c: Mor ×_Ob Mor → Mor defined in D, all satisfying the usual category axioms expressed as commutative diagrams in D. If D is Set, this data yields an ordinary small category. If D carries additional algebraic structure (for example D = Ab, the category of abelian groups), the resulting internal category encodes a category object in that environment, reflecting the ambient algebraic context.

Alternatively, a D-category can denote a category enriched over a monoidal category D. In this setting, for any pair of objects x and y, there is a hom-object Hom_D(x, y) in D, along with composition and identity maps given by morphisms in D: Hom_D(y, z) ⊗ Hom_D(x, y) → Hom_D(x, z) and I → Hom_D(x, x). When D is Set with the cartesian product, this recovers ordinary categories; if D is Ab, one obtains additive or linear categories, depending on the enrichment.

The term D-category is informal and overlaps with related notions such as internal categories, enriched categories, and fibered or indexed categories. Precision usually requires specifying whether one means an internal category in D or a category enriched over D.

See also enriched category, internal category, double category, fibered category.