*-autonomous category

In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object $\bot$ .

Definition

Let C be a symmetric monoidal closed category. For any object A and $\bot$ , there exists a morphism

\partial_{A,\bot}:A\to(A\Rightarrow\bot)\Rightarrow\bot

defined as the image by the bijection defining the monoidal closure, of the morphism

\mathrm{eval}_{A,A\Rightarrow\bot}\circ\gamma_{A\Rightarrow\bot,A} : (A\Rightarrow\bot)\otimes A\to\bot

An object $\bot$ of the category C is called dualizing when the associated morphism $\partial_{A,\bot}$ 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 $(-)^*:C^{\mathrm{op}}\to C$ such that for every object A there is a natural isomorphism $A\cong{A^*}^*$ , and for every three objects A, B and C there is a natural bijection

\mathrm{Hom}(A\otimes B,C^*)\cong\mathrm{Hom}(A,(B\otimes C)^*)

The dualizing object of C is then defined by $\bot=I^*$ .

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

A^*\otimes B^* \to (B\otimes A)^*

These are all isomorphisms if and only if the *-autonomous category is compact closed.

Examples

A familiar example is given by matrix theory as finite-dimensional linear algebra, namely 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.

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.

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 which 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.

References

Michael Barr (1979). Springer-Verlag, ed. "*-autonomous Categories". Lecture Notes in Mathematics 752.
Michael Barr (1995). "Non-symmetric *-autonomous Categories" (PDF). Theoretical Computer Science 139: 115–130. doi:10.1016/0304-3975(94)00089-2.
Michael Barr (1999). "*-autonomous categories: once more around the track" (PDF). Theory and Applications of Categories 6: 5–24.

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