Monads, or triples, and the algebras they define, proved to be very important in several fields of mathematics. For example, they allow us to see the algebraic nature owned by topological spaces. This fact, for what concerns compact Hausdorff spaces, is known since 1969 (see [4]). By suitably weakening the axioms of the algebras, M. Barr in 1970 (see [1]) generalized the result to all topological spaces. This step represented a starting point of the new theory of lax (Eilenberg-Moore) algebras, introduced about thirty years later by M.M. Clementino, D. Hofmann and W. Tholen (see [2] and [3]).
[1] M. Barr, Relational Algebras, in: Reports of the Midwest Category Seminar, IV, pp 39-55, Lecture Notes in Mathematics 137, Springer, Berlin (1970). [2] M.M. Clementino and D. Hofmann, Topological features of lax algebras, Appl. Categ. Structures 11 (2003), 267-286. [3] M.M. Clementino and W. Tholen, Metric, topology and multicategory - a common approach, J. Pure Appl. Algebra 179 (2003), 13-47. [4] E. Manes, A triple theoretic construction of compact algebras, 1969 Sem. on Triples and Categorical Homology Theory (ETH Zurich 1966/67), 91118, Lecture Notes in Mathematics, Springer, Berlin.
(*) Pier Giorgio Basile is a student for the Joint PhD Program in Mathematics UC|UP working at University of Coimbra in the area of "Algebra, Logic and Topology" under the supervision of Prof. Maria M. Clementino. The seminar takes place in PORTO.
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