Free Algebras in Varieties

We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to one of equational classes defined by equation arrows. Free algebras in the varieties are investigated and their existence is proved under the assumptions of accessibility. In universal algebra we deal with varieties – classes of algebras satisfying a certain collection of identities (pairs of terms of the corresponding language). This concept was generalized by Adámek and Porst in [3]. They worked with algebras for an endofunctor on a cocomplete category and used the free-algebra construction developed by Adámek in [1] (a chain of term-functors) to define equation arrows as certain regular epimorphisms. Using them they defined equational categories as analogues to varieties. These categories are later studied in [5]. We focus on another approach to varieties of algebras for a functor. We also use the free-algebra construction and define a natural term as a natural transformation with codomain in a term-functor. A pair of natural terms with a common domain will be called a natural identity and will be satisfied on an algebra if both of its natural transformations have the same term-evaluation on this algebra. Natural identities induce classes of algebras, which are proved to be precisely the classes defined by means of equation arrows. We present several examples of such classes and show that, in some cases, this approach essentially simplifies the presentation. In the second chapter we investigate free algebras in a variety. Induction by natural identities allows us to make a restriction on identities with domains preserving the colimits of some small chains. Such identities will be called accessible. These cases still cover most of the usual examples and we prove that such varieties have free algebras. The proof uses a conversion of variety to a category of algebras for a diagram of monads used by Kelly in [7] to define algebraic colimit of monads. His theorem proving the existence of free objects of this category yields the existence of free algebras in the variety induced by accessible identities. Notational convention. The constant functor mapping the objects to object X will be denoted by CX . The initial object in a cocomplete category will be denoted by 0. For functors, we omit the brackets and the composition mark ◦ when possible. The class of objects and morphisms of a category will be denoted by Ob and Mor, 2000 Mathematics Subject Classification: primary 08C05; secondary 18C05, 18C20.