Compact closed category

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.
Symmetric compact closed category.
A symmetric monoidal category formula_1 is compact closed if every object formula_2 has a dual object. If this holds, the dual object is unique up to canonical isomorphism, and is denoted formula_3. 
In a bit more detail, an object formula_3 is called the dual of formula_5 if it is equipped with two morphisms called the unit formula_6 and the counit formula_7, satisfying the equations
and
where formula_10 are the introduction of the unit on the left and right, respectively, and formula_11 is the associator.
For clarity, we rewrite the above compositions diagrammatically. In order for formula_1 to be compact closed, we need the following composites to equal formula_13:
and formula_15:
Definition.
More generally, suppose formula_1 is a monoidal category, not necessarily symmetric, such as in the case of a pregroup grammar. The above notion of having a dual formula_3 for each object "A" is replaced by that of having both a left and a right adjoint, formula_19 and formula_20, with a corresponding left unit formula_21, right unit formula_22, left counit formula_23, and right counit formula_24. These must satisfy the four yanking conditions, each of which are identities:
and
That is, in the general case, a compact closed category is both left and right-rigid, and biclosed.
Non-symmetric compact closed categories find applications in linguistics, in the area of categorial grammars and specifically in pregroup grammars, where the distinct left and right adjoints are required to capture word-order in sentences. In this context, compact closed monoidal categories are called (Lambek) pregroups.
Properties.
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 formula_29, one can define
which can be shown to be a proper trace. It helps to draw this diagrammatically:
formula_31
Examples.
The canonical example is the category FdVect with finite-dimensional vector spaces as objects and linear maps as morphisms. Here formula_3 is the usual dual of the vector space formula_5.
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.
Simplex category.
The simplex category can be used to construct an example of non-symmetric compact closed category. The simplex category is the category of non-zero finite ordinals (viewed as totally ordered sets); its morphisms are order-preserving (monotone) maps. We make it into a monoidal category by moving to the arrow category, so the objects are morphisms of the original category, and the morphisms are commuting squares. Then the tensor product of the arrow category is the original composition operator. The left and right adjoints are the min and max operators; specifically, for a monotone map "f" one has the right adjoint
and the left adjoint 
The left and right units and counits are:
One of the yanking conditions is then
The others follow similarly. The correspondence can be made clearer by writing the arrow formula_41 instead of formula_42, and using formula_43 for function composition formula_44.
Dagger compact category.
A dagger symmetric monoidal category which is compact closed is a dagger compact category.
Rigid 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.