*-autonomous category

In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object formula_1. The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality.
Definition.
Let C be a symmetric monoidal closed category. For any object "A" and formula_1, there exists a morphism
defined as the image by the bijection defining the monoidal closure
of the morphism
where formula_6 is the "symmetry" of the tensor product. An object formula_1 of the category C is called dualizing when the associated morphism formula_8 is an isomorphism for every object "A" of the category C. 
Equivalently, a *-autonomous category is a symmetric monoidal category "C" together with a functor formula_9 such that for every object "A" there is a natural isomorphism formula_10, and for every three objects "A", "B" and "C" there is a natural bijection
The dualizing object of "C" is then defined by formula_12. The equivalence of the two definitions is shown by identifying formula_13.
Properties.
Compact closed categories are *-autonomous, with the monoidal unit as the dualizing object. Conversely, if the unit of a *-autonomous category is a dualizing object then there is a canonical family of maps
These are all isomorphisms if and only if the *-autonomous category is compact closed.
Examples.
A familiar example is the category of finite-dimensional vector spaces over any field "k" made monoidal with the usual tensor product of vector spaces. The dualizing object is "k", the one-dimensional vector space, and dualization corresponds to transposition. Although the category of all vector spaces over "k" is not *-autonomous, suitable extensions to categories of topological vector spaces can be made *-autonomous. 
On the other hand, the category of topological vector spaces contains an extremely wide full subcategory, the category Ste of stereotype spaces, which is a *-autonomous category with the dualizing object formula_15 and the tensor product formula_16. 
Various models of linear logic form *-autonomous categories, the earliest of which was Jean-Yves Girard's category of coherence spaces.
The category of complete semilattices with morphisms preserving all joins but not necessarily meets is *-autonomous with dualizer the chain of two elements. A degenerate example (all homsets of cardinality at most one) is given by any Boolean algebra (as a partially ordered set) made monoidal using conjunction for the tensor product and taking 0 as the dualizing object.
The formalism of Verdier duality gives further examples of *-autonomous categories. For example, mention that the bounded derived category of constructible l-adic sheaves on an algebraic variety has this property. Further examples include derived categories of constructible sheaves on various kinds of topological spaces.
An example of a self-dual category that is not *-autonomous is finite linear orders and continuous functions, which has * but is not autonomous: its dualizing object is the two-element chain but there is no tensor product.
The category of sets and their partial injections is self-dual because the converse of the latter is again a partial injection.
The concept of *-autonomous category was introduced by Michael Barr in 1979 in a monograph with that title. Barr defined the notion for the more general situation of "V"-categories, categories enriched in a symmetric monoidal or autonomous category "V". The definition above specializes Barr's definition to the case "V" = Set of ordinary categories, those whose homobjects form sets (of morphisms). Barr's monograph includes an appendix by his student Po-Hsiang Chu that develops the details of a construction due to Barr showing the existence of nontrivial *-autonomous "V"-categories for all symmetric monoidal categories "V" with pullbacks, whose objects became known a decade later as Chu spaces.
Non symmetric case.
In a biclosed monoidal category "C", not necessarily symmetric, it is still possible to define a dualizing object and then define a *-autonomous category as a biclosed monoidal category with a dualizing object. They are equivalent definitions, as in the symmetric case.