Linton's theorem (equational theory)

Linton's theorem concerns the correspondence between monad and Lawvere theories. These two notions are the two main category theoretic formulations of universal algebra.[1] Although these results are not explicitly stated, they were essentially introduced by Linton (1969). The following (2) and (3) were first explicitly stated and proven by Dubuc (1970, Theorem III.), even in the more general setting of enriched category.[2] The fact that a monad is a category theoretic formulation of universal algebra is not intuitive, given its historical origins. In actually, monads originated in algebraic topology and typically arise from a pair of adjoint functors.[1] However, there is the following result concerning between monads and universal algebra, which is attributed to Linton: the category of models of a Lawvere theory is equivalent to the categories of Eilenberg–Moore algebras of the monad corresponding to that Lawvere theory.

Modern statement of Linton's theorem

  1. The category of finitary monads on is equivalent to the category of Lawvere theories and, subject to a generalization in the definition of Lawvere theory, every monad arises thus, uniquely up to coherent isomorphism.[1]
  2. Infinitary Lawvere theories are equivalent to monads on .[2]
  3. The category of models of an infinitary Lawvere theory is equivalent to the category of algebras of its monad.[2]

Note

References

  • Hyland, Martin; Power, John (April 2007). "The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads". Electronic Notes in Theoretical Computer Science. 172: 437–458. doi:10.1016/j.entcs.2007.02.019.
  • Bourke, John; Garner, Richard (July 2019). "Monads and theories". Advances in Mathematics. 351: 1024–1071. arXiv:1805.04346. doi:10.1016/j.aim.2019.05.016.
  • Linton, F. E. J. (1966). "Some Aspects of Equational Categories". Proceedings of the Conference on Categorical Algebra. pp. 84–94. doi:10.1007/978-3-642-99902-4_3. ISBN 978-3-642-99904-8.
  • Linton, F. E. J. (1969). "An outline of functorial semantics". Seminar on Triples and Categorical Homology Theory. Lecture Notes in Mathematics. Vol. 80. pp. 7–52. doi:10.1007/BFb0083080. ISBN 978-3-540-04601-1.
  • Lucyshyn-Wright, Rory B. B. (2016). "Enriched algebraic theories and monads for a system of arities". Theory and Applications of Categories. 31: 101–137. doi:10.70930/tac/uidgtsv3.
  • Dubuc, Eduardo J. (1970). "Enriched semantics-structure (meta) adjointness" (PDF). Revista de la Unión Matemática Argentina. 25: 5–26.
  • Nishizawa, Koki; Power, John (March 2009). "Lawvere theories enriched over a general base". Journal of Pure and Applied Algebra. 213 (3): 377–386. doi:10.1016/j.jpaa.2008.07.009.
  • Adámek, J.; Rosický, J.; Vitale, E. M.; Lawvere, F. W. (18 November 2010). "Appendix A - Monads". Algebraic Theories: A Categorical Introduction to General Algebra. doi:10.1017/CBO9780511760754.026.
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