A category in which each object has a dual is called autonomous or rigid.
Consider an object in a monoidal category . The object is called a left dual of if there exist two morphsims
- , called the coevaluation, and , called the evaluation,
such that the following two diagrams commute
The object is called the right dual of . Left duals are canonically isomorphic when they exist, as are right duals. When C is braided (or symmetric), every left dual is also a right dual, and vice versa.
Categories with duals
A monoidal category where every object has a left (resp. right) dual is sometimes called a left (resp. right) autonomous category. Algebraic geometers call it a left (resp. right) rigid category. A monoidal category where every object has both a left and a right dual is called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.
- Peter Freyd and David Yetter (1989). "Braided Compact Closed Categories with Applications to Low-Dimensional Topology". Advances in Mathematics 77 (2): 156–182. doi:10.1016/0001-8708(89)90018-2.
- André Joyal and Ross Street. "The Geometry of Tensor calculus II". Synthese Lib 259: 29–68.
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