Skip to main content Accessibility help

Gabriel–Ulmer duality and Lawvere theories enriched over a general base

  • STEPHEN LACK (a1) and JOHN POWER (a2)


Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set. We generalise that relationship from Set to an arbitrary locally presentable category such as Poset and ωCpo or functor categories such as [Inj, Set] and [Inj, ωCpo]. That involves allowing the arities of Lawvere theories to be extended to being size-restricted objects of the locally presentable category. We develop a body of theory at this level of generality, in particular explaining how the relationship between generalised Lawvere theories and monads extends Gabriel–Ulmer duality.



Hide All
Adámek, J. & Rosický, J. (1994) Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, vol. 189. Cambridge University Press.
Barr, M. & Wells, C. (1985) Toposes, Triples and Theories. Springer.
Barr, M. & Wells, C. (1990) Category Theory for Computing Science. Prentice Hall.
Benton, N., Hughes, J. & Moggi, E. (2002) Monads and effects. In Advanced Lectures from International Summer School on Applied Semantics, APPSEM 2000 (Caminha, September 2000), Barthe, G., Dybjer, P., Pinto, L. & Saraiva, J. (eds), Lecture Notes in Computer Science, vol. 2395. Springer, pp. 42122.
Heckmann, R. (1994) Probabilistic domains. In Proceedings of the 19th Internatinal Colloquium in Trees in Algebra and Programming, CAAP '94 (Edinburgh, April 1994), Tison, S. (ed), Lecture Notes in Computer Science, vol. 136. Springer, pp. 2156.
Hyland, M., Levy, P. B., Plotkin, G. & Power, J. (2007) Combining algebraic effects with continuations, Theor. Comput. Sci., 375 (1–3): 2040.
Hyland, M., Plotkin, G. & Power, J. (2006) Combining computational effects: sum and tensor, Theor. Comput. Sci., 357 (1–3): 7099.
Hyland, M. & Power, J. (2006) Discrete Lawvere theories and computational effects, Theor. Comput. Sci., 366 (1–2): 144162.
Hyland, M. & Power, J. (2007) The category-theoretic understanding of universal algebra: Lawvere theories and monads. In Computation, Meaning, and Logic: Articles Dedicated to Gordon Plotkin, Cardelli, L., Fiore, M. & Winskel, G. (eds), Electronic Notes in Theoretical Computer Science, vol. 172. Elsevier, pp. 437458.
Joyal, A. & Street, R. (1993) Pullbacks equivalent to pseudo-pullbacks, Cahiers Topol. Géom. Différ. Catég., 34 (2): 153156.
Kelly, G. M. (1982a) Basic Concepts of Enriched Category Theory. Cambridge University Press.
Kelly, G. M. (1982b) Structures defined by finite limits in the enriched context I, Cahiers Topol. Géom. Différ. Catég., 23 (1): 342.
Lawvere, F. W. (1963) Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. USA, 50 (5): 869872.
Lüth, C. & Ghani, N. (2002) Monads and modularity. In Proceedings of the 4th International Workshop on Frontiers of Combining Systems, FroCoS 2002 (Santa Margherita Ligure, April 2002), Armando, A. (ed), Lecture Notes in Artificial Intellegince, vol. 2309. Springer, pp. 1832.
Moggi, E. (1989) Computational lambda-calculus and monads. In Proceedings of the 4th Annual IEEE Symposium on Logic in Computer Science, LICS '89 (Pacific Grove, CA, June 1989). IEEE CS Press, pp. 1423.
Moggi, E. (1991) Notions of computation and monads, Inform. Comput., 93 (1): 5592.
Nishizawa, K. & Power, J. (2009) Lawvere theories enriched over a general base, J. Pure Appl. Algebra, 213 (3): 377386.
O'Hearn, P. W. & Tennent, R. D. (1997) Algol-Like Languages. Birkhäuser.
Plotkin, G. & Power, J. (2002) Notions of computation determine monads. In Proceedings of the 5th International Confernce on Foundations of Software Science and Computation Structures, FOSSACS 2002 (Grenoble, April 2002), Nielsen, M. & Engberg, U. (eds), Lecture Notes in Computer Science, vol. 2303. Springer, pp. 342356.
Plotkin, G. & Power, J. (2003) Algebraic operations and generic effects, Appl. Categ. Struct., 11 (1): 6994.
Power, J. (1995) Why tricategories? Inform. Comput., 120 (2): 251262.
Power, J. (2000) Enriched Lawvere theories, Theory Appl. Categ., 6: 8393.
Power, J. (2006) Semantics for local computational effects. In Proceedings of the 22nd Annual Conference on Mathematical Foundations of Programming Semantics, MFPS-XXII (Genova, May 2006), Brookes, S. & Mislove, M. (eds), Electronic Notes in Theoretical Computer Science, vol. 158. Elsevier, pp. 355371.
Robinson, E. (2002) Variations on algebra: monadicity and generalisations of equational theories, Formal Aspects Comput., 13 (3–5): 308326.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

Gabriel–Ulmer duality and Lawvere theories enriched over a general base

  • STEPHEN LACK (a1) and JOHN POWER (a2)
Submit a response


No Discussions have been published for this article.


Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *