×

zbMATH — the first resource for mathematics

Diagram chasing in Mal’cev categories. (English) Zbl 0722.18005
Summary: A diagram chasing technique generalizing the ‘two-square’ lemma of homological algebras is extended from Mal’cev varieties to Mal’cev categories: regular categories in which all reflexive relations are effective. The principal method used is the calculus of relations. The connection with Goursat’s theorem in group theory is discussed.

MSC:
18G50 Nonabelian homological algebra (category-theoretic aspects)
18C05 Equational categories
08B05 Equational logic, Mal’tsev conditions
18E10 Abelian categories, Grothendieck categories
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barr, M.; Grillet, P.A.; van Osdal, D.H., Exact categories and categories of sheaves, () · Zbl 0223.18009
[2] Carboni, A., Categories of affine spaces, J. pure appl. algebra, 61, 243-250, (1989) · Zbl 0683.18008
[3] Carboni, A.; Walters, R.F.C., Cartesian bicategories I, J. pure appl. algebra, 49, 11-32, (1987) · Zbl 0637.18003
[4] Faro, E., On a conjecture of lawvere, ()
[5] Findlay, G.O., Reflexive homomorphic relations, Canad. math. bull., 3, 131-132, (1960) · Zbl 0100.28002
[6] Feyd, P., On canonizing category theory, or, on functorializing model theory, ()
[7] Goursat, É., Sur LES substitutions orthogonales…, Ann. sci. école norm. sup., 6, 3, 9-102, (1889) · JFM 21.0673.03
[8] Grothendieck, A., Sur quelques points d’algèbre homologique, Tôhoku math. J., 9, 119-221, (1957) · Zbl 0118.26104
[9] Hagemann, J.; Mitschke, A., On n-permutable congruences, Algebra universalis, 3, 8-12, (1973) · Zbl 0273.08001
[10] Lambek, J., Goursat’s theorem and the Zassenhaus lemma, Canad. J. math., 10, 45-56, (1957) · Zbl 0079.25202
[11] Lambek, J., Goursat’s theorem and homological algebra, Canad. math. bull., 7, 597-608, (1964) · Zbl 0124.01601
[12] Lambek, J., Lectures on rings and modules, (1966), Blaisdell Waltham, Chelsea, New York, 1976, 1986 · Zbl 0143.26403
[13] J. Lambek, On the ubiquity of Mal’cev operations, Contemp. Math., to appear. · Zbl 0770.08003
[14] Leicht, J.B., Axiomatic proof of J. Lambek’s homological theorem, Canad. math. bull., 7, 609-613, (1964) · Zbl 0124.01602
[15] Mal’cev, A.I., On the general theory of algebraic systems, Mat. sb. (N.S.), 35, 3-20, (1954) · Zbl 0057.02403
[16] Meisen, J., Relations in categories, () · Zbl 0331.18007
[17] Meisen, J., Relations in regular categories, (), 90-92
[18] Smith, J.D.H., Mal’cev varieties, () · Zbl 0344.08002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.