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.
[1] Often, an object is dualizable only when it satisfies some finiteness or compactness property.
[2] A category in which each object has a dual is called autonomous or rigid.
This expression makes sense in any category with an appropriate replacement for the tensor product of vector spaces.
For any monoidal category (C, ⊗) one may attempt to define a dual of an object V to be an object V∗ ∈ C with a natural isomorphism of bifunctors For a well-behaved notion of duality, this map should be not only natural in the sense of category theory, but also respect the monoidal structure in some way.
[1] An actual definition of a dual object is thus more complicated.
In a closed monoidal category C, i.e. a monoidal category with an internal Hom functor, an alternative approach is to simulate the standard definition of a dual vector space as a space of functionals.
Algebraic geometers call it a left (respectively right) rigid 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.