nLab dagger-compact category

Idea

A $\dagger$-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 $\dagger$-adjoint.

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

Example

(finite-dimensional 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.

The finite parts of quantum mechanics (quantum information theory and quantum computation) are naturally formulated as the theory of $\dagger$-compact categories. For more on this see at finite quantum mechanics in terms of †-compact categories.

Example

(spans)
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 $I \stackrel{!}{\leftarrow} X \stackrel{Id \times Id}{\to} X \times X$. See Baez 2007.

Properties

Relation to self-duality

If each object $X$ of a compact closed category is equipped with a self-duality structure $X \simeq X^\ast$, then sending morphisms to their dual morphisms but with these identifications pre- and postcomposed

$$(-)^\dagger \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (Y \stackrel{\simeq}{\to} Y^\ast \stackrel{f^\ast}{\longrightarrow} X^\ast \stackrel{\simeq}{\to} X)$$

constitutes a dagger-compact category structure.

See for instance (Selinger, remark 4.5).

Applied for instance to the category of finite-dimensional inner product spaces this dagger-operation sends matrices to their transpose?.

A good example of a $\dagger$-compact category where most objects are not isomorphic to their duals is the category of continuous unitary representations of U(n) on finite-dimensional complex Hilbert spaces.

Related concepts

• bra-ket notation

• quantum information theory via dagger-compact categories

References

The concept was introduced in

• Samson Abramsky, 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:quant-ph/0402130)

with an expanded version in

• Samson Abramsky, Bob Coecke, Categorical quantum mechanics, in Handbook of Quantum Logic and Quantum Structures, Elsevier (2008) [arXiv:0808.1023, ISBN:9780080931661]

under the name “strongly compact” and used for finite quantum mechanics in terms of dagger-compact categories. The topic was taken up

• 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)

where the alternative terminology “dagger-compact” was proposed, and used for the abstract characterization of quantum operations (completely positive maps on Bloch regions of density matrices).

The examples induced from self-duality-structure are discussed abstractly in

• Peter Selinger, Autonomous categories in which $A \simeq A^\ast$, talk at QPL 2010 (pdf)

That finite-dimensional Hilbert spaces are “complete for dagger-compactness” is shown in

• Peter Selinger (2012), Finite dimensional Hilbert spaces are complete for dagger compact closed categories, (arXiv:1207.6972).

The example of spans:

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