Dual object

In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite dimensional vector spaces. An object admitting a dual is called a dualizable object. In this formalism, infinite dimensional vector spaces are not dualizable, since the dual vector space V∗ doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property. A category in which each object has a dual is called autonomous or rigid. The category of finite dimensional vector spaces with the standard tensor product is rigid, while the category of all vector spaces is not.

Consider a monoidal category (VectK, ⊗K) of vector spaces over a field K with the standard tensor product. A space V is dualizable if and only if it is finite dimensional, and in this case the dual object V∗ coincides with the standard notion of a dual vector space. Consider a monoidal category (ModR, ⊗R) of modules over a commutative ring R with the standard tensor product. A module M is dualizable if and only if it is a finitely generated projective module. In that case the dual object M∗ is also given by the module of homomorphisms HomR(M, R). Consider a homotopy category of pointed spectra Ho(Sp) with the smash product as the monoidal structure. If M is a compact neighborhood retract in (for example, a compact smooth manifold), then the corresponding pointed spectrum Σ∞(M+) is dualizable. This is a consequence of Spanier–Whitehead duality, which implies in particular Poincaré duality for compact manifolds.

A monoidal category where every object has a left (respectively right) dual is sometimes called a left (respectively right) autonomous category. Algebraic geometers call it a left (respectively right) rigid category. A monoidal category where every object has both a left and a right dual is called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.

Any endomorphism f of a dualizable object admits a trace, which is a certain endomorphism of the monoidal unit of C. This notion includes, as very special cases, the trace in linear algebra and the Euler characteristic of a chain complex.