# Normalizers, centralizers and action accessibility

## 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.

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