# nLab dagger-compact category

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Idea

A $\dagger$-compact category is a category which is a

and a

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

• every object has a dual;

• every morphism has an $\dagger$-adjoint.

(Hence a $\dagger$-compact category is similar in flavor to an $(\infty,2)$-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 $\dagger$-compact if the dagger-operation takes units of dual objects to counits in that for every object $A$ of $C$ we have

$\array{ && A \otimes A^* \\ & {}^{\epsilon_A^\dagger}\nearrow \\ I && \downarrow^{\mathrlap{\sigma_{A \times A^*}}} \\ & {}_{\eta_A}\searrow \\ && A^* \otimes A } \,.$

## Examples

• For $C$ a category with finite limits the category $Span_1(C)$ whose morphisms are spans in $C$ is $\dagger$-compact. The $\dagger$ 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 $X \stackrel{Id}{\leftarrow} X \stackrel{Id \times Id}{\to} X \times X$. See

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

## Quantum mechanics in terms of $\dagger$-compact categories

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

For more on this see

## Relation to Hilbert spaces

The category of Hilbert spaces (over the complex numbers) with finite dimension is a standard example of a $\dagger$-compact category. This example is complete? for equations in the language of $\dagger$-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)