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Algebraically coherent categories

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Alan S. Cigoli, James R. A. Gray and Tim Van der Linden

We call a finitely complete category *algebraically coherent* if
the change-of-base functors of its fibration of points are
*coherent*, which means that they preserve finite limits and
jointly strongly epimorphic pairs of arrows. We give examples of
categories satisfying this condition; for instance, coherent
categories, *categories of interest* in the sense of Orzech, and
(compact) Hausdorff algebras over a semi-abelian algebraically
coherent theory. We study equivalent conditions in the context of
semi-abelian categories, as well as some of its consequences:
including amongst others, strong protomodularity, and normality of
Higgins commutators for normal subobjects, and in the varietal case,
fibre-wise algebraic cartesian closedness.

Keywords:
Coherent functor; Smith, Huq, Higgins commutator; semi-abelian,
locally algebraically cartesian closed category; category of interest;
compact Hausdorff algebra

2010 MSC:
20F12, 08C05, 17A99, 18B25, 18G50}

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 54, pp 1864-1905.

Published 2015-12-08.

http://www.tac.mta.ca/tac/volumes/30/54/30-54.pdf

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