# *-autonomous category

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

## Definition

Let C be a symmetric monoidal closed category, for any object A and ${\displaystyle \bot }$, there exists a morphism

${\displaystyle \partial _{A,\bot }:A\to (A\Rightarrow \bot )\Rightarrow \bot }$

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

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

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

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

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

## 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

${\displaystyle A^{*}\otimes B^{*}\to (B\otimes A)^{*}}$ .

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.