Extension theory of Maltsev algebras
by
Jonathan D. H. Smith
Department of Mathematics, Iowa State University, Ames, IA50011,

Because of the recent attention given to the homological algebra of non-abelian structures, and in the interests of minimizing the duplication of known results, it appears timely to recall the extension theory of Mal’tsev algebras [2].

Work in a variety T of Mal’tsev algebras. (One may readily expand to the context of a Mal’tsev category by the standard translations, e.g. from congruence to internal reflexive relation.) A simplicial object B truncated at dimension 2 is said to be seeded if surjectivity conditions hold, and if, for eⁱ: B² --> B¹, the kernel congruence of e⁰ is the centralizer of the kernel of e¹. Seeded simplicial objects correspond to Barr'’s clas sE [1]. A congruence a on a T-object A is said to plant the seeded simplicial object a^(\eta(a)|a) ===> A^\eta(a) --> A^{a o \eta(a)}.

Theorem. Given a seeded simplicial object B and (regular) epi p⁰ onto B⁰, there is a congruence a on a T-object A planting B, with p⁰ at the bottom, if and only if p⁰ is unobstructed.

As usual, the group of (equivalence classes of) singular extensions under Baer sum acts regularly on the group of non-singular extensions. Obstructions are classified by a higher cohomology group.

References

[1] B. Eckmann (ed.), “Seminar on Triples and Categorical Homology Theory, ”Springer Lecture Notes in Mathematics No. 80, 1969.

[2] J. D. H. Smith, “Ma'l’cev Varieties, ” Springer Lecture Notes in Mathematics No. 554, 1976.

Date received: May 6, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajf-04.