internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
Kleisli triple is the term chosen in Moggi (1991), Def. 1.2 for what was previously called an algebraic theory in extension form in Manes (1976), Ex. 3.12 (p. 32) and what later authors refer to as extension systems.
All of these terms refer to the equivalent axiomatization of monads (old terminology: “triples”) via the ingredients of their Kleisli categories under the Kleisli equivalence.
This is the natural and common incarnation of monads in computer science; see there for more.
Eugenio Moggi, Def. 1.2 in: Computational lambda-calculus and monads, in: Proceedings of the Fourth Annual Symposium on Logic in Computer Science (1989) 14-23 [doi:10.1109/LICS.1989.39155]
Eugenio Moggi, Def. 1.2 in: Notions of computation and monads, Information and Computation, 93 1 (1991) [doi:10.1016/0890-5401(91)90052-4, pdf]
following
Lecture notes:
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