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.

Relation to Hilbert spaces

The category of Hilbert spaces (over the complex numbers) with finite dimension is a standard example of a $†$-compact category. This example is complete? for equations in the language of $†$-compact categories; see Selinger 2012.

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 (2007), 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)

For completeness of finite-dimensional Hilbert spaces:

• Peter Selinger (2012), Finite dimensional Hilbert spaces are complete for dagger compact closed categories, arXiv.