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.
Keywords: Frobenius algebras, linear distributive categories
2000 MSC: 03F52,18D10,18D15
Theory and Applications of Categories,
Vol. 24, 2010,
No. 2, pp 25-38.
http://www.tac.mta.ca/tac/volumes/24/2/24-02.dvi
http://www.tac.mta.ca/tac/volumes/24/2/24-02.ps
http://www.tac.mta.ca/tac/volumes/24/2/24-02.pdf
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