×

Models of Horn theories. (English) Zbl 0676.03022

Categories in computer science and logic, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Colo. 1987, Contemp. Math. 92, 1-7 (1989).
[For the entire collection see Zbl 0667.00009.]
In model theory logical theories are used to present categories of models. A useful and different approach is also provided by the theory of sketches and the category of models of a sketch. The notion of a sketch was developed by C. Ehresmann and much work has been done by the French school of category theorists (in particular Guitart and Lair). A lucid and highly readable account of sketches appears in the book of M. Barr and C. Wells [Toposes, triples and theories (1985; Zbl 0567.18001)]. This paper explores the relationship between equational Horn theories and finite limit (FL) sketches and their categories of models. The category of models of an equational Horn theory is always sketchable by a FL sketch, but the converse need not hold. This leads to the main result of the paper, which states that a category C which is sketchable by a FL sketch is the category of models of a generalized equational Horn theory iff C has a regular generating class of regular projectives. Some related aspects of this result are then discussed.
Reviewer: K.Rosenthal

MSC:

03C05 Equational classes, universal algebra in model theory
18C05 Equational categories
08B30 Injectives, projectives