Abstract
Let k be a regular infinite cardinal number. A primitive (=equationally definable) class of algebras is called k-ary provided the algebras can be described in a specified way by means of operations whose arity is less than k. Such k-ary primitive classes of algebras (resp. k-ary varietal categories) have been characterized categorically by Lawvere [5] in case k = ℵo and by Linton [7], Felscher [1], and others in the general case. In this paper we propose a general definition of k-arity for arbitrary concrete categories, show that this concept is useful at least for algebraic categories, defined below, and coincides with the corresponding concept for varietal categories.
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References
FELSCHER, W.: Kennzeichnung von primitiven und quasiprimitiven Kategorien von Algebren. Archiv Math.19, 390–397 (1968).
HERRLICH, H.: Factorizations of morphisms f:B→FA, Math. Zeitschr.114, 180–186 (1970).
ISBELL, J.: Subobjects, adequacy, completeness and categories of algebras. Rozprawy Mat.36, 1–32 (1964).
— Normal completions of categories. Lecture Notes47, 110–155 (1967).
LAWVERE, F. W.: Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. USA50, 869–872 (1963).
LENZING, H.: Über die Funktoren Ext1(-,E) und Tor-1(-,E). Dissertation Berlin (1964).
LINTON, F. E. J.: Some aspects of equational categories, Proc. Conf. Categorical Algebra. La Jolla 1965, Berlin, Heidelberg 1966, 84–94.
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Herrlich, H. A characterization of k-ary algebraic categories. Manuscripta Math 4, 277–284 (1971). https://doi.org/10.1007/BF01190281
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DOI: https://doi.org/10.1007/BF01190281