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[email protected] Department of Computer Science Research Report No. RR-02-01 ISSN 1470-5559 February 2002 Variations on Algebra: monadicity and generalisations of equational theories Edmund Robinson Variations on Algebra: monadicity and generalisations of equational theories Edmund Robinson ∗ Queen Mary, University of London February 12, 2002 Dedicated to Rod Burstall Introduction Rod Burstall gave me my first academic job. This was, he explained, to help his group learn about some of the more advanced aspects of category theory. In fact I spent much of my time learning from them about some of the more advanced, now fundamental, aspects of Computer Science. I’ve always been grateful to Rod, both for the opportunity, and for his willingness to give me time to learn. So it’s appropriate that I offer this essentially tutorial paper to him. Better late than never! Back in the dawn of time Linton [Lin66] discovered that there was a connection between one- sorted algebraic theories and the categorical notion of monad, or more precisely, monads with rank on the category of sets. From the categorist’s point of view this is important because it gives a good ranking of the definitional power of the notion of monad, placing monads at the bottom end of a hierarchy of theories with generic models, with geometric theories and their classifying toposes at the top.