nLab
dagger-compact category

Idea

A $†$-compact category is a category which is a

• compact closed category

and a

• symmetric monoidal †-category

in a compatible way. So, notably, it is a monoidal category in which

• every object has a dual;

• every morphism has an $†$-adjoint.

(Hence a $†$-compact category is similar in flavor to an $(\mathrm{\infty },2)$-category with all adjoints in the sense of On the Classification of Topological Field Theories .)

Definition

A category $C$ that is equipped with the structure of a symmetric monoidal †-category and is compact closed is $†$-compact if the dagger-operation takes units of dual objects to counits in that for every object $A$ of $C$ we have

$$\begin{array}{ccc}& & A\otimes A^{*}\\ & {}^{{ϵ}_A^{†}}\nearrow \\ I& & {\downarrow }^{{\sigma }_{A\times A^{*}}}\\ & {}_{{\eta }_A}\searrow \\ & & A^{*}\otimes A\end{array}.$$\array{ && A \otimes A^* \\ & {}^{\epsilon_A^\dagger}\nearrow \\ I && \downarrow^{\mathrlap{\sigma_{A \times A^*}}} \\ & {}_{\eta_A}\searrow \\ && A^* \otimes A } \,.

Examples

• For $C$ a category with finite limits the category ${\mathrm{Span}}_1(C)$ whose morphisms are spans in $C$ is $†$-compact. The $†$ operation is that of relabeling the legs of a span as source and target. The tensor product is defined using the cartesian product in $C$. Every object $X$ is dual to itself with the unit and counit given by the span $X\stackrel{\mathrm{Id}}{\leftarrow }X\stackrel{\mathrm{Id}\times \mathrm{Id}}{\to }X\times X$. See

  • John Baez, Spans in quantum theory (web, pdf, blog)

Quantum mechanics in terms of $†$-compact categories

Large parts of quantum mechanics and quantum computation are naturally formulated as the theory of $†$-compact categories.

For more on this see

• quantum mechanics in terms of †-compact categories.

References

The concept was introduced in

• Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols, in Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04), IEEE Computer Science Press, 2004. (arXiv)

See also:

• Peter Selinger, Dagger compact closed categories and completely positive maps, in Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), ENTCS 170 (2007), 139–163. (web, pdf)