Category theoretic understandings of universal algebra and its dual: monads and Lawvere theories, comonads and what?

Reference:

Behrisch, M., Kerkhoff, S. and Power, J., 2012. Category theoretic understandings of universal algebra and its dual: monads and Lawvere theories, comonads and what? Electronic Notes in Theoretical Computer Science, 286, pp. 5-16.

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Official URL:

http://dx.doi.org/10.1016/j.entcs.2012.08.002

Abstract

Universal algebra is often known within computer science in the guise of algebraic specification or equational logic. In 1963, it was given a category theoretic characterisation in terms of what are now called Lawvere theories. Unlike operations and equations, a Lawvere theory is uniquely determined by its category of models. Except for a caveat about nullary operations, the notion of Lawvere theory is equivalent to the universal algebraistʼs notion of an abstract clone. Lawvere theories were soon followed by a further characterisation of universal algebra in terms of monads, the latter quickly becoming preferred by category theorists but not by universal algebraists. In the 1990ʼs began a systematic attempt to dualise the situation. The notion of monad dualises to that of comonad, providing a framework for studying transition systems in particular. Constructs in universal algebra have begun to be dualised too, with different leading examples. But there is not yet a definitive dual of the concept of Lawvere theory, or that of abstract clone, or even a definitive dual of operations and equations. We explore the situation here.

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