### Duality

The Frobenius equations mentioned for example in the last post were introduced in my paper with Carboni in 1987 to express the fact that an object V of a symmetric monoidal category might be self dual - that is, that V is adjoint to V - in a strong sense. Each object of the category of sets and relations is self-dual in the strong sense that it has a Frobenius algebra structure (given by the diagonal map; that is, arising from equality). Even the Frobenius structure of wires in electrical circuits mentioned in the last post has an idea of equality associated, in that case the wires are equipotential regions.

However categories are not self-dual objects in the category of categories and bimodules (=profunctors); the natural dual of a category, the opposite category, is not isomorphic to the category. Here I describe a generalization of the notion of Frobenius algebra which implies that an object has a dual, but is not necessarily self-dual, and which applies to the category of categories and bimodules.

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However categories are not self-dual objects in the category of categories and bimodules (=profunctors); the natural dual of a category, the opposite category, is not isomorphic to the category. Here I describe a generalization of the notion of Frobenius algebra which implies that an object has a dual, but is not necessarily self-dual, and which applies to the category of categories and bimodules.

Read more »

Labels: category theory, Como Category Seminar, mathematics