nLab
dagger-compact category

Context

Monoidal categories

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.

Definition

A category CC 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 AA of CC we have

AA * ϵ A I σ A×A * η A A *A. \array{ && A \otimes A^* \\ & {}^{\epsilon_A^\dagger}\nearrow \\ I && \downarrow^{\mathrlap{\sigma_{A \times A^*}}} \\ & {}_{\eta_A}\searrow \\ && A^* \otimes A } \,.

Examples

Properties

Relation to self-duality

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

() :(XfY)(YY *f *X *X) (-)^\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 transposed matrix?.

References

The concept was introduced in

with an expanded version in

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 AA *A \simeq A^\ast, talk at QPL 2010 (pdf)

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

Revised on October 25, 2016 08:02:51 by Anonymous Coward (130.225.121.207)