*-autonomous category

March 2023 Symmetric monoidal closed category equipped with a dualizing object In mathematics, a *-autonomous (read "star-autonomous") category is a symmetric monoidal closed category equipped with a dualizing object . The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality. Definition Let be a symmetric monoidal closed category . For any pair of objects, in particular A and , there exists a morphism defined as the image by the bijection defining the monoidal closure of the evaluation map: where is the symmetry of the tensor product. An object of the category is called dualizing when the associated morphism is an isomorphism for every object A of . Equivalently, a *-autonomous category is a symmetric monoidal category 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 is then defined by . The equivalence of the two definitions is shown by identifying . 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 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 (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, Boyarchenko Drinfeld 2013 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 , 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. See also Autonomous category References Michael Barr (mathematician) Michael Barr 1979 *-Autonomous Categories Lecture Notes in Mathematics 752 Springer 10.1007/BFb0064579 978-3-540-09563-7 Michael Barr 14721961 1995 Non-symmetric *-autonomous Categories Theoretical Computer Science 139 115–130 10.1016/0304-3975(94)00089-2 Michael Barr 1999 *-autonomous categories: once more around the track Theory and Applications of Categories 6 http://www.tac.mta.ca/tac/volumes/6/n1/n1.pdf 5–24 Boyarchenko Mitya Drinfeld Vladimir Vladimir Drinfeld A duality formalism in the spirit of Grothendieck and Verdier Quantum Topology 4 2013 4 447–489 3134025 10.4171/QT/45 1108.6020 55605535 star-autonomous+category star-autonomous category Category:Monoidal categories Category:Closed categories Category:Duality (mathematics)