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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.
@article{TAC_2010_24_a1,
author = {J.M. Egger},
title = {The {Frobenius} relations meet linear distributivity},
journal = {Theory and applications of categories},
pages = {25--38},
publisher = {mathdoc},
volume = {24},
year = {2010},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2010_24_a1/}
}
J.M. Egger. The Frobenius relations meet linear distributivity. Theory and applications of categories, Tome 24 (2010), pp. 25-38. http://geodesic.mathdoc.fr/item/TAC_2010_24_a1/