Paper Info
Title
Category Theoretic Understandings of Universal Algebra and its Dual: Monads and Lawvere Theories, Comonads and What?
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 algebraist@?s 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 1990@?s 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.
Year
DOI
Venue
2012
10.1016/j.entcs.2012.08.002
Electr. Notes Theor. Comput. Sci.
Keywords
Field
DocType
universal algebra,computer science,abstract clone,category theoretic characterisation,category theorist,category theoretic understandings,lawvere theories,lawvere theory,universal algebraist,algebraic specification,universal algebraists,different leading example,monad,comonad
Lawvere theory,Algebraic specification,Discrete mathematics,Monad (category theory),Comma category,2-category,Equational logic,Universal algebra,Mathematics,Monad (functional programming)
Journal
Volume
ISSN
Citations 
286,
1571-0661
2
PageRank 
References 
Authors
0.48
8
3
Name
Order
Citations
PageRank
Mike Behrisch1196.90
Sebastian Kerkhoff2225.93
John Power3777.79