# *-autonomous category

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

## Definition[edit]

Let **C** be a symmetric monoidal 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 category *C* 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 .

## Properties[edit]

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[edit]

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[edit]

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[edit]

- Michael Barr (1979). Springer-Verlag, ed. "*-autonomous Categories".
*Lecture Notes in Mathematics*.**752**. doi:10.1007/BFb0064579. - 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. - star-autonomous+category in
*nLab*