Abstract
Our aim is to analyze and to publicize two interesting properties — well known in universal algebra for varieties — that a regular category, and in particular an exact category, may possess: theMaltsev property, asserting the permutabilitySR=RS of equivalence relations on any object, and the weakerGoursat property, asserting only thatSRS=RSR. We investigate these properties, give various equivalent forms of them, and develop some of their useful consequences.
Similar content being viewed by others
References
M. Barr: Catégories exactes,C. R. Acad. Sci. Paris. Sér. A–B 272 (1971), A1501-A1503.
M. Barr, P.A. Grillet, and D.H. van Osdol: Exact categories and categories of sheaves,Lecture Notes in Mathematics 236, Springer, Berlin (1971).
M. Barr: On categories with effective unions,Lecture Notes in Mathematics 1384, Springer-Verlag (1988), 19–35.
M. Barr and C. Wells:Toposes, Triples, and Theories, Springer-Verlag, New York-Heidelberg-Berlin-Tokyo, 1985.
A. Carboni, J. Lambek, and M.C. Pedicchio: Diagram chasing in Mal'cev categories,J. Pure Appl. Algebra 69 (1991), 271–284.
A. Carboni and S. Mantovani: An elementary characterization of categories of separated objects,J. Pure Appl. Algebra 89 (1993), 63–92.
A. Day and R. Freese: A characterization of identities implying congruence modularity I,Canad. J. Math. 32 (1980), 1140–1167.
B.J. Day and G.M. Kelly: On topological quotient maps preserved by pullbacks or products,Proc. Cambridge Phil. Soc. 67 (1970), 553–558.
E. Faro: On a conjecture of Lawvere, Preprint, SUNY Buffalo, 1989.
T.H. Fay: On commuting congruences in regular categories,Math. Coll. Univ. Cape Town 11 (1977), 13–31.
T.H. Fay: On categorial conditions for congruences to commute,Algebra Univ. 8 (1978), 173–179.
P.J. Freyd and A. Scedrov:Categories, Allegories, North-Holland, Amsterdam-New York-Oxford-Tokyo, 1990.
N. Funuyama and T. Nakayama: On the distributivity of a lattice of lattice congruences,Proc. Imp. Acad. Sci. Tokyo 18 (1942), 553–554.
P. Gabriel and M. Zisman:Calculus of Fractions and Homotopy Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1967.
É. Goursat: Sur les substitutions orthogonales ...,Ann. Sci. Éc. Norm. Sup. 3(6) (1889), 9–102.
J. Hagemann and A. Mitschke: Onn-permutable congruences,Algebra Univ. 3 (1973), 8–12.
G. Janelidze and G.M. Kelly: Galois theory and a general notion of central extension,J. Pure Appl. Algebra, to appear.
P.T. Johnstone:Stone Spaces, Cambridge University Press, 1982.
P.T. Johnstone: Affine categories and naturally Mal'cev categories,J. Pure Appl. Algebra 61 (1981), 251–256.
G.M. Kelly: Monomorphisms, epimorphisms, and pull-backs,J. Austral. Math. Soc. 9 (1969), 124–142.
G.M. Kelly: A note on relations relative to a factorization system,Proc. Conf. on Category Theory (Como, 1990),Springer Lecture Notes in Mathematics 1448 (1991), 249–261.
A Klein: Relations in categories,Ill. J. Math. 14 (1970), 536–550.
J. Lambek: Goursat's theorem and homological algebra,Can. Math. Bull. 7 (1964), 597–608.
J. Lambek: On the ubiquity of Mal'cev operations,Contemporary Math. 131 (1992), 135–146.
A.I. Mal'cev: On the general theory of algebraic systems,Mat. Sbornik N.S. 35 (1954), 3–20.
J. Meisen:Relations in Categories, Thesis, McGill Univ., 1972.
J. Meisen: On bicategories of relations and pullback spans,Comm. Alg. 1 (1974), 377–401.
A. Mitschke: Implication algebras are 3-permutable and 2-distributive,Algebra Univ. 1 (1971), 182–186.
J.C. Moore: Homotopie des complexes monoïdaux, Séminaire H. Cartan, 1954/55, Exposé 18.
G. Richter: Mal'cev conditions for categories, inCategorical Topology (Proc. Conference Toledo, Ohio 1983), Heldermann Verlag, Berlin 1984, pp. 453–469.
C.M. Ringel: The intersection property of amalgamations,J. Pure Appl. Algebra 2 (1972), 341–342.
E.T. Schmidt: Onn-permutable equational classes,Acta Sci. Math. (Szeged) 33 (1972), 29–30.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Carboni, A., Kelly, G.M. & Pedicchio, M.C. Some remarks on Maltsev and Goursat categories. Appl Categor Struct 1, 385–421 (1993). https://doi.org/10.1007/BF00872942
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00872942