View source for *-Autonomous category

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Let $\mathcal{C}$ be a symmetric [[Closed monoidal category|closed monoidal category]] (cf. also [[Category|Category]]). A [[Functor|functor]] $( - ) ^ { * } : \cal C ^ { \operatorname{op} } \rightarrow C$ is a duality functor if there exists an isomorphism $d ( A , B ) : B ^ { A } \overset{\cong}{\rightarrow} A ^ { * } B ^ { * }$, natural in $A$ and $B$, such that for all objects $A , B , C \in \mathcal{C}$ the following diagram commutes:

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300108.png"/></td> </tr></table>

where in the bottom arrow $s = s ( ( A ^ { * } ) ^ { ( B ^ { * } ) } , ( B ^ { * } ) ^ { ( C ^ { * } ) } )$.

A category is $*$-autonomous if it is a symmetric monoidal closed category with a given duality functor.

It so happens that $*$-autonomous categories have real-life applications: they are models of (at least the finite part of) linear logic [[#References|[a2]]] and have uses in modelling processes.

An example of a $*$-autonomous category is the category $\mathcal{R} \text{el}$ of sets and relations; duality is given by $S ^ { * } = S$. In fact, $B ^ { A } \cong ( A ^ { * } \otimes B )$.

From a given symmetric monoidal closed category and an object in it (that serves as a dualizing object) one can construct a $*$-autonomous category (the so-called Chu construction, [[#References|[a3]]]). It can be viewed as a kind of generalized topology.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Barr,   "$*$-Autonomous categories" , ''Lecture Notes in Mathematics'' , '''752''' , Springer  (1979)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Barr,   C. Wells,   "Category theory for computing science" , Publ. CRM  (1990)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P.-H. Chu,   "Constructing $*$-autonomous categories"  M. Barr (ed.) , ''$*$-Autonomous categories'' , ''Lecture Notes in Mathematics'' , '''752''' , Springer  (1979)  pp. Appendix</td></tr></table>