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.

A symmetric monoidal category ${\displaystyle (\mathbf {C} ,\otimes ,I)}$ is compact closed if every object ${\displaystyle A\in \mathbf {C} }$ has a dual object. If this holds, the dual object is unique up to canonical isomorphism, and is denoted ${\displaystyle A^{*}}$.

In a bit more detail, an object ${\displaystyle A^{*}}$ is called the dual of ${\displaystyle A}$ if it is equipped with two morphisms called the unit ${\displaystyle \eta _{A}:I\to A^{*}\otimes A}$ and the counit ${\displaystyle \varepsilon _{A}:A\otimes A^{*}\to I}$, satisfying the equations

and

where ${\displaystyle \lambda ,\rho }$ are the introduction of the unit on the left and right, respectively, and ${\displaystyle \alpha }$ is the associator.

For clarity, we rewrite the above compositions diagrammatically. In order for ${\displaystyle (\mathbf {C} ,\otimes ,I)}$ to be compact closed, we need the following composites to equal ${\displaystyle \mathrm {id} _{A}}$:

and ${\displaystyle \mathrm {id} _{A^{*}}}$:

More generally, suppose ${\displaystyle (\mathbf {C} ,\otimes ,I)}$ is a monoidal category, not necessarily symmetric, such as in the case of a pregroup grammar. The above notion of having a dual ${\displaystyle A^{*}}$ for each object A is replaced by that of having both a left and a right adjoint, ${\displaystyle A^{l}}$ and ${\displaystyle A^{r}}$, with a corresponding left unit ${\displaystyle \eta _{A}^{l}:I\to A\otimes A^{l}}$, right unit ${\displaystyle \eta _{A}^{r}:I\to A^{r}\otimes A}$, left counit ${\displaystyle \varepsilon _{A}^{l}:A^{l}\otimes A\to I}$, and right counit ${\displaystyle \varepsilon _{A}^{r}:A\otimes A^{r}\to I}$. These must satisfy the four yanking conditions, each of which are identities: