Perfect categories

This note extends to categories Fountain's theorem (2) that for a perfect monoid S, every flat S-set is projective. (The converse is known (4).)

[1]  T. M. Bisgaard.  On Perfect Semigroups , 1976, Proceedings of the Edinburgh Mathematical Society.

[2]  B. Mitchell.  The dominion of Isbell , 1972 .

[3]  B. Stenström.  Flatness and localization over monoids , 1971 .