TY - JOUR

T1 - Category theoretic understandings of universal algebra and its dual: monads and Lawvere theories, comonads and what?

AU - Behrisch, Mike

AU - Kerkhoff, Sebastian

AU - Power, John

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84879196184&partnerID=8YFLogxK

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

U2 - 10.1016/j.entcs.2012.08.002

DO - 10.1016/j.entcs.2012.08.002

M3 - Article

VL - 286

SP - 5

EP - 16

JO - Electronic Notes in Theoretical Computer Science

JF - Electronic Notes in Theoretical Computer Science

SN - 1571-0661

T2 - 28th Conference on the Mathematical Foundations of Programming Semantics

Y2 - 6 June 2012 through 9 June 2012

ER -