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Relative internal actions

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James Richard Andrew Gray and Tamar Janelidze-Gray

For a relative exact homological category (C,E), we define
*relative points* over an arbitrary object in C, and show that
they form an exact homological category. In particular, it follows that
the full subcategory of nilpotent objects in an exact homological category
is an exact homological category. These nilpotent objects are defined with
respect to a Birkhoff subcategory in C as defined by T. Everaert and T.
Van der Linden. In addition, we
introduce *relative internal actions* and show that, just as in the
classical case, there is an equivalence of categories between the category
of relative points over an object and the category of relative internal
actions for the same object.

Keywords:
relative semi-abelian category, relative homological category,
semi-abelian category, homological category, category of relative points,
relative internal actions

2010 MSC:
18G50, 18G25, 18A20, 18A05, 18A25, 18A32, 18C20

*Theory and Applications of Categories,*
Vol. 31, 2016,
No. 16, pp 444-461.

Published 2016-05-30.

http://www.tac.mta.ca/tac/volumes/31/16/31-16.pdf

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