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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300102.png" /> be a symmetric [[Closed monoidal category|closed monoidal category]] (cf. also [[Category|Category]]). A [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300103.png" /> is a duality functor if there exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300104.png" />, natural in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300106.png" />, such that for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300107.png" /> the following diagram commutes:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300108.png" /></td> </tr></table>
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where in the bottom arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a1300109.png" />.
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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:
  
A category is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001011.png" />-autonomous if it is a symmetric monoidal closed category with a given duality functor.
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<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>
  
It so happens that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001012.png" />-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.
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where in the bottom arrow $s = s ( ( A ^ { * } ) ^ { ( B ^ { * } ) } , ( B ^ { * } ) ^ { ( C ^ { * } ) } )$.
  
An example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001013.png" />-autonomous category is the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001014.png" /> of sets and relations; duality is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001015.png" />. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001016.png" />.
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A category is $*$-autonomous if it is a symmetric monoidal closed category with a given duality functor.
  
From a given symmetric monoidal closed category and an object in it (that serves as a dualizing object) one can construct a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001017.png" />-autonomous category (the so-called Chu construction, [[#References|[a3]]]). It can be viewed as a kind of generalized topology.
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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.
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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, $B ^ { A } \cong ( A ^ { * } \otimes B )$.
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001018.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001019.png" />-autonomous categories"  M. Barr (ed.) , ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130010/a13001020.png" />-Autonomous categories'' , ''Lecture Notes in Mathematics'' , '''752''' , Springer  (1979)  pp. Appendix</TD></TR></table>
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<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>

Revision as of 17:01, 1 July 2020

Let $\mathcal{C}$ be a symmetric closed monoidal category (cf. also Category). A 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:

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 [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, [a3]). It can be viewed as a kind of generalized topology.

References

[a1] M. Barr, "$*$-Autonomous categories" , Lecture Notes in Mathematics , 752 , Springer (1979)
[a2] M. Barr, C. Wells, "Category theory for computing science" , Publ. CRM (1990)
[a3] P.-H. Chu, "Constructing $*$-autonomous categories" M. Barr (ed.) , $*$-Autonomous categories , Lecture Notes in Mathematics , 752 , Springer (1979) pp. Appendix
How to Cite This Entry:
*-Autonomous category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=*-Autonomous_category&oldid=12194
This article was adapted from an original article by Michel Eytan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article