Compact closed category

Special kind of category with "dual objects"
In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite dimensional vector space. So, the motivating example of a compact closed category is FdVect, the category having finite dimensional vector spaces as objects and linear maps as morphisms, with tensor product as the monoidal structure. Another example is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure.

Compact closed categories are a special case of monoidal closed categories, which in turn are a special case of closed categories. Compact closed categories are precisely the symmetric autonomous categories. They are also * autonomous. Every compact closed category C admits a trace. Namely, for every morphism , one can define

which can be shown to be a proper trace. It helps to draw this diagrammatically:

The canonical example is the category FdVect with finite dimensional vector spaces as objects and linear maps as morphisms. Here is the usual dual of the vector space
The category of finite dimensional representations of any group is also compact closed. The category Vect, with all vector spaces as objects and linear maps as morphisms, is not compact closed; it is symmetric monoidal closed.

A dagger symmetric monoidal category which is compact closed is a dagger compact category.

A monoidal category that is not symmetric, but otherwise obeys the duality axioms above, is known as a rigid category. A monoidal category where every object has a left (resp. right) dual is also sometimes called a left (resp. right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is then a compact closed category.