Category Theory
If A is a finitary algebraic category, then the forgetful functor
is monadic. Even more, it is strictly monadic, in that the comparison functor
is an isomorphism (and not just an equivalence). Here, is the Eilenberg–Moore category on . In general, one says a category is an algebraic category if it is monadic over . This is a more general notion than "finitary algebraic category" (the notion of "variety" used in universal algebra) because it admits such categories as CABA (complete atomic Boolean algebras) and CSLat (complete semilattices) whose signatures include infinitary operations. In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded arity. The algebraic category of sigma algebras also has infinitary operations, but their arity is countable whence its signature is small (forms a set).
Read more about this topic: Variety (universal Algebra)
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