category with duals (list of them)
dualizable object (what they have)
A -compact category is a category which is a
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 -category with all adjoints in the sense of On the Classification of Topological Field Theories .)
A category 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 of we have
For a category with finite limits the category whose morphisms are spans in 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 . Every object is dual to itself with the unit and counit given by the span . See
For more on this see
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.
The concept was introduced in
For completeness of finite-dimensional Hilbert spaces: