A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids

Abstract

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an additional motivation for developing commutator theory. On the other hand, commutator theory suggests: (a) another notion of central extension that turns out to be equivalent to the Galois-theoretic one under surprisingly mild additional conditions; (b) a way to describe internal groupoids in ‘nice’ categories. This is essentially a 20 year story (with only a couple of new observations), from introducing categorical Galois theory in 1984 by the author, to obtaining and publishing final forms of results (a) and (b) in 2004 by M. Gran and by D. Bourn and M. Gran, respectively.

Keywords

• Galois theory
• Galois structure
• internal groupoid
• central extension
• commutator
• Mal'tsev category
• congruence modularity
• congruence permutability
• shifting property