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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Knots, groups, cospans and spans III

This post is the third of a series, the first being In praise of composition: knot groups and cospans and the second Knots, groups, cospans and spans.

I want to give some details,  and a slight extension to the remarks of the second post. In brief I will show how the Frobenius equations characterize the existence of inverses, and how the tensor product of Frobenius objects gives a semi-direct product of groups.

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Mathematical economics - double-entry bookkeeping

I want to talk a little about a mathematical description of accounting rather than economics in general. When I was making the move from Australia to Italy, which I finally did in 1998, I had some particular trouble understanding my financial resources. I thought about the method of keeping accounts introduced in Italy, and first described by Luca Pacioli in 1494, called double-entry bookkeeping or partita doppia.

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Knots, groups, cospans and spans

In the last few days I have clarified the ideas described in the last post (partially responding to objections by Aurelio Carboni), introducing further structure which John Armstrong does not seem to have noticed.

The clarification which I will describe makes evident that the two invariants described by John Armstrong (i) the fundamental group of (the complement of) a knot, and (ii) the number of colourings of a knot, are both examples of a single phenomenon, and both can be extended beyond tangles.

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