Rosický, Jiří Elementary categories. (English) Zbl 0665.18004 Arch. Math. 52, No. 3, 284-288 (1989). The paper contributes to the problem of characterizing categories of models of usual one-sorted finitary first-order theories. Any locally finitely presentable category with a finitely presentable generator is of this kind. Surprisingly, the same is true for any category of many-sorted sets. (This result was further developed by G. Jarzembski [Finitary fibrations, Cah. Topologie Géom. Differ. (to appear)].) Finally, Bankston’s conjecture is proved (independently of B. Banaschewski [Can. J. Math. 36, 1113-1118 (1984; Zbl 0561.18004)]). Reviewer: J.Rosický Cited in 1 ReviewCited in 5 Documents MSC: 18A99 General theory of categories and functors 03C68 Other classical first-order model theory 18B99 Special categories Keywords:compact Hausdorff space; categories of models; one-sorted finitary first- order theories; locally finitely presentable category; finitely presentable generator; category of many-sorted sets Citations:Zbl 0561.18004 PDFBibTeX XMLCite \textit{J. Rosický}, Arch. Math. 52, No. 3, 284--288 (1989; Zbl 0665.18004) Full Text: DOI References: [1] B. Banaschewski, More on compact Hausdorff spaces and finitary duality. Canad. J. Math.36, 1113-1118 (1984). · Zbl 0561.18004 · doi:10.4153/CJM-1984-063-6 [2] B. Banaschewski andH. Herrlich, Subcategories defined by implications. Houston J. Math.2, 149-171 (1976). · Zbl 0344.18002 [3] P. Bankston, Some obstacles to duality in topological algebras. Canad. J. Math.34, 80-90 (1982). · Zbl 0476.18006 · doi:10.4153/CJM-1982-008-6 [4] C. C.Chang and H. J.Keisler, Model Theory. Amsterdam-New York-Oxford 1973. [5] M.Coste, Une approche logique des th?ories d?ffinissables par limites projectives finies. S?minaire J. B?nabou, Universit? Paris-Nord 1976. [6] M. A.Dickman, Large infinitary languages. Amsterdam 1975. [7] P.Gabriel and F.Ulmer, Lokalpr?sentierbare Kategorien. LNM221, Berlin-Heidelberg-New York 1971. · Zbl 0225.18004 [8] G. Jarzembski, Finitary fibration. Inst. Math. Copernicus Univ., Toru? 1986. [9] G. Jarzembski, Concrete categories representable by universally axiomatizable classes of finitary relational systems. Inst. Math. Copernicus Univ., Toru? 1987. [10] O. Keane, Abstract Horn theories. In: Model theory and topoi. LNM445, 15-50, Berlin-Heidelberg-New York 1973. [11] M. Makkai andR. Par?, Accessible categories: The foundations of categorical model theory. Dept. Math. McGill Univ., Montreal 1987. [12] M. Makkai andA. M. Pitts, Some results on locally finitely presentable categories. Trans. Amer. Math. Soc.229, 473-496 (1987). · Zbl 0615.18002 · doi:10.1090/S0002-9947-1987-0869216-2 [13] M. Richter, Limites in Kategorien von Relationsystemen. Z. Math. Logik. Grundlag. Math.17, 75-90 (1971). · Zbl 0227.02033 · doi:10.1002/malq.19710170114 [14] J. Rosick?, Categories of models. In: Proc. Category Theory Conf. Oberwolfach 1983, Seminarberichte Fernuniversit?t Hagen 2,19, 377-413 (1984). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.