Tensored

Tensored is a term used in category theory and related areas to describe a category or construction that admits a tensoring operation with objects from a fixed monoidal category. In its general form, a category C is said to be tensored over a monoidal category V if for every object X in C and every object V in V there is an object X ⊗ V in C, called the tensor of X by V.

The tensoring is required to satisfy a natural universal property: there are natural isomorphisms Hom_C(X ⊗ V, Y) ≅ Hom_V(V, Hom_C(X, Y)) that are natural in Y for all objects X in C and V, and Y in C. If V has a unit object I, there is also a unit isomorphism X ⊗ I ≅ X for all X in C. Coherence conditions give associativity up to isomorphism: (X ⊗ V) ⊗ W ≅ X ⊗ (V ⊗ W). When these data exist, C is said to be tensored over V; the dual notion is cotensored, which uses an internal hom-like object rather than a tensor.

Concrete examples help grounding the concept. The category of modules over a ring R, Mod-R, can be viewed as tensored over the category of abelian groups Ab via the tensor product over the integers: for M ∈ Mod-R and A ∈ Ab, one obtains M ⊗_Z A ∈ Mod-R. In a broader sense, many abelian or enriched categories carry a tensoring with Ab, or with their underlying monoidal category, enabling constructions that combine objects of C with parameter objects from V.

Tensoredness is a central notion in enriched category theory, where it formalizes how a category interacts with a chosen monoidal context, and it often pairs with cotensoredness to support a full range of limits and colimits.