Action accessibility via centralizers

Cigoli, Alan Stefano;Mantovani, Sandra
(2012) Journal of Pure and Applied Algebra —

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  • Cigoli, Alan Stefanoorcid-logoUCLouvain
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  • Mantovani, Sandra
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Abstract
In this paper we first introduce a non-symmetric notion of centralization between a relation S and an equivalence relation R, which coincides with Smith centralization in the case S is an equivalence relation too. We then prove that in any action accessible category in the sense of Bourn and Janelidze (2009) [11], the centralizer of an equivalence relation R, defined as in [11], actually has a stronger property, namely it is an equivalence relation, which is the largest among all the relations S centralizing R in the non-symmetric sense mentioned above. As a main result, we show that the existence of centralizers for any equivalence relation with this stronger property actually characterizes action accessibility for exact protomodular categories.
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Cigoli, A. S., & Mantovani, S. (2012). Action accessibility via centralizers. Journal of Pure and Applied Algebra. https://doi.org/10.1016/j.jpaa.2012.02.023