Abstract
The notions of a locally α-presentable and locally α-generated categories are introduced, where α is a regular cardinal. The properties of these categories are studied extensively, in particular their close relationship with other types of categories. Also the subclasses of topos, algebraic categories and locally α-noetherian categories are investigated in detail. A “classification” of locally α-presentable, locally α-generated categories, locally α-noetherian categories and algebraic categories is given.
This note is a summary of a joint paper with P. Gabriel. It is an outgrowth of mostly unpublished papers and manuscripts of both authors from 1965–69, cf. [8], [20], [21], [22]. Details will appear elsewhere.
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Bibliography
J. Benabou. Structures algébriques dans les catégories, thèses, Univérsité de Paris, 1966.
G. Birkhoff. On the structure of abstract algebras, Proc. Cambridge Phil. Soc. vol. 31, 1935.
S. Breitsprecher. Lokal endlich präsentierbare Kategorien. Math. Seminar, Universität Giessen, 1970.
M. Bunge. Characterization of diagrammatic categories, Dissertation, University of Pennsylvannia, 1966.
S. Eilenberg, J. Moore. Adjoint functors and triples, Ill. J. Math. 9, 1965.
P. Freyd. Algebra valued functors in general categories and tensor products in particular, Colloq. Math. vol. 14, 1966.
P. Gabriel. Des catégories abéliennes, Bull. Soc. Math. de France, vol. 92, 1962.
P. Gabriel. Rétracts et catégories algébriques, handwritten draft, 1966.
J. W. Gray. Sheaves with values in arbitrary categories, Topology, vol. 3, 1965.
A. Grothendieck. Sur quelques points d'algèbre homologique, Töhoko, Math. J. 9, 1957.
P. Hilton. On the category of direct systems and functors on groups, J. of Pure and Applied Algebra, vol. 1, 1971.
J. Isbell. Small adequate subcategories, Ill. J. Math. 4, 1960.
F. W. Lawvere. Functorial semantics and algebraic theories, Proc. Nat. Acad. Sci. USA 50, 1963.
F. W. Lawvere-Tierney. The elementary theory of abstract sheaves, to appear.
F. Linton. Some aspects of equational categories, coca (La Jolla) Springer, 1966.
Morita. Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo, Kyoiku Daigaku, Sec. 6, 1958.
H. Reichel und Kapphengst. Algebraische Theorien und Kan'sche Erweiterungen, to appear.
J. Roos. Comptes rendus 259, 1964, p. 970.
Slominski. The theory of abstract algebras with infinitary operations, Rozprawy Mat. 18, 1959.
F. Ulmer. Properties of Kan extensions, mimeographed notes, ETH, 1966.
F. Ulmer. Properties of dense and relative adjoints, J. of Algebra 8, 1968.
F. Ulmer. Triples in algebraic categories, mimeographed notes, ETH, 1969.
J. Verdier. Séminaire de géometrie algébrique, fascicule 1, Inst. Hautes Études Scient., 1963/64.
H. Wolff. Fractions and closed categories, Dissertation, University of Illinois, 1970.
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Ulmer, F. (1971). Locally α-presentable and locally α-generated categories. In: Gray, J.W. (eds) Reports of the Midwest Category Seminar V. Lecture Notes in Mathematics, vol 195. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072314
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DOI: https://doi.org/10.1007/BFb0072314
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