Autonomous category
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In mathematics, an autonomous category is a monoidal category where dual objects exist.[1]
[edit] Definition
A left (resp. right) autonomous category is a monoidal category where every object has a left (resp. right) dual. An autonomous category is a monoidal category where every object has both a left and a right dual.[2] In this sense, autonomous categories are also known as rigid categories.
In a symmetric monoidal category, the existence of left duals is equivalent to the existence of right duals, categories of this kind are called compact closed categories.
The concepts of *-autonomous category and autonomous category are not directly related, however, any symmetric autonomous category (that is, any compact closed category) is *-autonomous.
[edit] Notes and references
- ^ Some authors use this term for a symmetric monoidal closed category, or for a biclosed monoidal category when symmetry is not assumed.
- ^ Berman, pp 34
[edit] Sources
- Yetter, David N. (2001). Functorial Knot Theory. World Scientific. ISBN 9810244436.
- Berman, Stephen; Yuly Billi (2003). Vertex Operator Algebras in Mathematics and Physics. American Mathematical Society. ISBN 0821828568.
