With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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.
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
(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.
(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.
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
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.
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
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
That finite-dimensional Hilbert spaces are “complete for dagger-compactness” is shown in
The example of spans:
Last revised on September 12, 2023 at 06:58:47. See the history of this page for a list of all contributions to it.