In category theory, a tensor category (ℭ, ⊗, I, a, r, ℓ) consists of a category ℭ, an object I of ℭ, a functor ⊗:ℭ×ℭ->ℭ, and a natural isomorphism a | = | a_(U V W) :(U⊗V)⊗W->U⊗(V⊗W) r | = | r_V :V⊗I->V ℓ | = | ℓ_V :I⊗V->V, where the data are subject to the following axioms: 1. Given four objects U, V, W, and X of ℭ, the top diagram above commutes. 2. Given two objects U and V of ℭ, the bottom diagram above commutes.

category | category theory | functor | morphism | natural isomorphism | natural transformation | object | strict tensor category | unital natural transformation