The topic of the present chapter is varieties and quasi varieties, i.e., equational and implicational classes of algebras (with a set of many-sorted operations). Whereas locally presentable categories are characterized as categories of models of limit sketches, varieties can be characterized as categories of models of product sketches. These are limit sketches with discrete diagrams; in fact, to sketch a given variety of algebras, the Lawvere-Linton algebraic theory of that variety, considered as a product-sketch, can be used (Theorems 3.16 and 3.30). Moreover, varieties are precisely the accessibly monadic categories over many-sorted sets (Theorem 3.31). Quasi varieties can be abstractly characterized as precisely the locally presentable categories with a dense set of regular projectives; and varieties are then characterized as precisely the quasivarieties with effective equivalence relations (Theorem 3.33).

The name “presentable” stems from algebra: an algebra is a λ-presentable object of a finitary variety iff it can be presented by less than λ generators and less than λ equations in the usual algebraic sense [Theorem 3.12].

The chapter is concluded by a characterization of locally presentable categories which, although known as folklore, has never been published before: locally presentable categories are precisely the essentially algebraic categories, i.e., varieties of partial algebras in which the domain of definition of each partial operation is described by equations involving total operations only (Theorem 3.36).

Since all the results in the present chapter are quite analogous in the finitary case and in the general case, we present all the details for the (notationally simpler) finitary algebras, and then mention the general results more briefly.