Dualfactor

Dualfactor is a term used in some mathematical contexts to denote a component of a factorization that corresponds, under a specified duality, to the other component. The idea is that in a setting with a duality functor, objects can be decomposed into complementary factors, and applying the duality to one factor yields a natural dual partner called the dualfactor of the other.

Formal definition: In a monoidal category with a duality functor D, if an object X admits a factorization X ≅ A ⊗ B, the dualfactor of A with respect to this decomposition is D(B). This yields a dual factorization X ≅ A ⊗ D(B) together with the canonical evaluation map ε: A ⊗ D(B) → I to the unit object.

Examples: In finite-dimensional vector spaces over a field with the usual tensor product and duality, if X ≅ U ⊗ V, then the dualfactor of U is V*, the dual space of V. The evaluation map ε: U ⊗ V* → k provides the standard nondegenerate pairing between U and V in this setting.

Properties and caveats: The notion of a dualfactor depends on the chosen duality and the particular factorization. Different factorizations of the same object can produce different dualfactors, and not all dualities yield meaningful dualfactors in every context. The construction is governed by coherence conditions in the underlying category, such as associativity and naturality of the duality.

See also: duality, factorization, dual module, evaluation map, monoidal category. The term dualfactor is not universally standardized and may be used with varying specifics in different mathematical disciplines.