TY - JOUR
ID - 15430
TI - A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Janelidze, George
AD - Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa
Y1 - 2016
PY - 2016
VL - 5
IS - 1
SP - 1
EP - 54
KW - Galois theory
KW - Galois structure
KW - internal groupoid
KW - central extension
KW - commutator
KW - Mal'tsev category
KW - congruence modularity
KW - congruence permutability
KW - shifting property
DO -
N2 - 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.
UR - https://cgasa.sbu.ac.ir/article_15430.html
L1 - https://cgasa.sbu.ac.ir/article_15430_1878ca09e32e895f11fc7b28a8efcb83.pdf
ER -