# *-Autonomous category

Let be a symmetric closed monoidal category (cf. also Category). A functor is a duality functor if there exists an isomorphism , natural in and , such that for all objects the following diagram commutes:

where in the bottom arrow .

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 of sets and relations; duality is given by . In fact, .

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. Michel Eytan (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=*-Autonomous_category&oldid=12194