Let C be a symmetricmonoidal closed category. For any object A and, there exists a morphism defined as the image by the bijection defining the monoidal closure of the morphism where is the symmetry of the tensor product. An object of the category C is called dualizing when the associated morphism is an isomorphism for every object A of the category C. Equivalently, a *-autonomous category is a symmetric monoidal categoryC together with a functor such that for every object A there is a natural isomorphism, and for every three objects A, B and C there is a natural bijection The dualizing object of C is then defined by. The equivalence of the two definitions is shown by identifying .
Properties
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 and the tensor product. 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 is given by any Boolean algebra 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 categoryV. The definition above specializes Barr's definition to the case V = Set of ordinary categories, those whose homobjects form sets. 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 categoriesV with pullbacks, whose objects became known a decade later as Chu spaces.
Non symmetric case
In a biclosed monoidal categoryC, 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.
