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Normalizers, centralizers and action accessibility

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J. R. A. Gray

We give several reformulations of action accessibility in the sense of D. Bourn
and G. Janelidze. In particular we prove that a pointed exact protomodular
category is action accessible if and only if for each normal monomorphism
$\kappa:X\to A$ the normalizer of $< \kappa,\kappa>: X\to A\times A$ exists.
This clarifies the connection between normalizers and action accessible
categories established in a joint paper of D. Bourn and the author, in which it
is proved that for pointed exact protomodular categories the existence of
normalizers implies action accessibility. In addition we prove a pointed exact
protomodular category with coequalizers is action accessible if centralizers of
normal monomorphisms exist, and the normality of unions holds.

Keywords:
action accessible, protomodular, Barr exact, normality, centrality,
normalizer, centralizer

2010 MSC:
18D35, 18A35, 18A05, 18A99

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 12, pp 410-432.

Published 2015-04-03.

http://www.tac.mta.ca/tac/volumes/30/12/30-12.pdf

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