The Frobenius relations meet linear distributivity
J.M. Egger · Theory and applications of categories · 2010
The notion of Frobenius algebra originally arose in ring theory, but it is a fairly easy observation that this notion can be extended to arbitrary monoidal categories.But, is this really the correct level of generalisation?For example, when studying Frobenius algebras in the * -autonomous category Sup, the standard concept using only the usual tensor product is less interesting than a similar one in which both the usual tensor product and its de Morgan dual (par ) are used.Thus we maintain that the notion of linear-distributive category (which has both a tensor and a par, but is nevertheless more general than the notion of monoidal category) provides the correct framework in which to interpret the concept of Frobenius algebra. Example. Every (planar) * -autonomous category (K, ×∩, e, -•, •-, d) has an underlying linearly distributive category, in which the second tensor product is defined as the de Morgan dual of the first.xHere, as usual, x * is an abbreviation for x -• d, and * x is an abbreviation for d •-x.