The aim of this note is to make the reader familiar with the notion of algebraic category. The approach we use is based on Lawvere’s pioneering work where the notions of algebraic theory and algebraic category are introduced as invariant formulations of Birkhoff’s universal algebra.
The beauty of the subject comes from the interplay between properties of the theory (the description, the syntax) and properties of the objects described (the models or semantics). In Section 1 we present these basic notions and prove the main properties of algebraic categories. One novelty in our approach in comparison to classical treatments on the subject is the use of reflexive graphs and of their coequalizers.
The fact that this kind of colimit is formed in an algebraic category as in the underlying category of sets, is of basic importance and will appear many times in the note. In Section 2 we discuss intensively Lawvere’s characterization of one-sorted algebraic categories and show that an algebraic category is a definite mathematical object, more precisely it is equivalent to an exact category with a set of regular projective, finitely presentable, regular generators.
In Section 3 we show how Lawvere’s Characterization Theorem is extremely useful when discussing applications to different important categorical situations. In particular, regular-epireflective subcategories of algebraic ones and quasi- algebraic categories will be completely characterized by weakening the hypothesis of the main Theorem.
Then, localizations of algebraic and quasi-algebraic categories will be investigated. To do that we will need sophisticated techniques based on the exact completion construction. Finally, the case of Mal’cev categories will also be discussed and characterized in a similar way.
In the last section, the idea of a model as a functor preserving a certain property will be generalized to the case of a locally finitely presentable category K = Lex(Cop,Set) where K is determined by finite-limit-preserving functors on a small category Cop with finite limits. Cop will be called the essentially algebraic theory of K.
Any algebraic category (or quasi-algebraic category) is locally finitely presentable with essentially algebraic theory the free equalizer completion of its algebraic theory. Conversely, there is no reason for a locally finitely presentable category to be algebraic (or quasi-algebraic). We will characterize these cases syntactically, i.e. in terms of conditions on the corresponding essentially algebraic theories.
It is important to notice that once more the results we give will be obtained as suitable application of Lawvere’s Characterization Theorem. In this approach the new notion of effective projective object will play a basic role.
