abelian group, free algebra, free module etc. It has the consequence that every algebra in a variety is a homomorphic image of a free algebra. Category theory Besides varieties, category theorists use two other frameworks that are equivalent in terms of the kinds of algebras they describe: finitary monads and Lawvere theories. We may go from a variety to a finitary monad as follows. A category with some variety of algebras as objects and homomorphisms as morphisms is called a finitary algebraic category. For any finitary algebraic category , the forgetful functor has a left adjoint , namely the functor that assigns to each set the free algebra on that set. This adjunction is monadic, meaning that the category is equivalent to the Eilenberg–Moore category for the monad . Moreover the monad is finitary, meaning it commutes with filtered colimits. The monad is thus enough to recover the finitary algebraic category. Indeed, finitary algebraic categories are precisely those categories equivalent to the Eilenberg-Moore categories of finitary monads. Both these, in turn, are equivalent to categories of algebras of Lawvere theories. Working with monads permits the following generalization. One says a category is an algebraic category if it is monadic over . This is a more general notion than "finitary algebraic category" 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). Every finitary algebraic category is a locally presentable category. Pseudovariety of finite algebras Since varieties are closed under arbitrary direct products, all non-trivial varieties contain infinite algebras. Attempts have been made to develop a finitary analogue of the theory of varieties. This led, e.g., to the notion of variety of finite semigroups. This kind of variety uses only finitary products. However, it uses a more general kind of identities. A pseudovariety is usually defined to be a class of algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. Not every author assumes that all algebras of a pseudovariety are finite; if this is the case, one sometimes talks of a variety of