Diagrams, equations and theories in categories. (English) Zbl 0547.18002

Universal algebra and its links with logic, algebra, combinatorics and computer science, Proc. 25. Arbeitstag. Allgemeine Algebra, Darmstadt 1983, Res. Expo. Math. 4, 143-149 (1984).
[For the entire collection see Zbl 0537.00003.]
This is an excellent short survey article on some links between Universal Algebra and Category Theory. The author concentrates on two important aspects. First, he states a metatheorem that, for a many-sorted algebraic theory \(\Omega\), all equations which are derived from reflexivity, symmetry and transitivity of equality, substitution of an equation in a term, and from substitution in an equation, correspond to derived commutative diagrams for internal \(\Omega\) -objects in a category C with products, and vice versa. This metatheorem simplifies, for instance, the study of category objects in a topos.
In the second part the author describes G. M. Kelly’s proof [Bull. Aust. Math. Soc. 26, 45-56 (1982; Zbl 0488.18001)] of a well-known result by A. Bastiani and Ch. Ehresmann [Cah. Topologie Géom. Différ. 13, 105-214 (1972; Zbl 0263.18009)] that sketches (S,\(\Phi)\) with a small category S and a set \(\Phi\) of cones in S can be completed to a sketch (T,\(\Psi)\) where \(\Psi\) consists of limiting cones such that a canonical equivalence of the corresponding categories of models arises through a generic (S,\(\Phi)\)-model in T.
Reviewer: W.Tholen


18C10 Theories (e.g., algebraic theories), structure, and semantics
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
08C05 Categories of algebras
18C05 Equational categories
18B25 Topoi
18D35 Structured objects in a category (MSC2010)
18-02 Research exposition (monographs, survey articles) pertaining to category theory