Algebraic Set Theory

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Why "algebraic set theory"?

There are two reasons for referring to this research as "algebraic set theory": The first reason is that the models of set theory that are produced by these methods are algebras for an abstractly presented "theory", in a precise, technical sense known to category theorists as a monad. The notion of an algebra for a monad subsumes and generalizes that of a model for a conventional algebraic theory, such as groups, rings, modules, etc. Indeed, the first significant work in this style on the applications of category theory to the study of set theory was the monograph Algebraic Set Theory (Cambridge, 1995) by André Joyal and Ieke Moerdijk. The second reason is that we believe the locution "algebraic logic" should properly refer to categorical logic rather than just the logic of Boole and his modern proponents, since categorical logic subsumes such lattice theoretic methods and not the other way around. Hence the term "algebraic set theory" rather than "categorical set theory". This is in keeping with the use of "algebraic" to mean, essentially, "functorial" in modern algebraic topology, algebraic geometry, etc.

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