#
A note on images of cover relations

##
J. R. A. Gray

For a category C, a
small category I, and a pre-cover relation c on C
we prove, under certain completeness assumptions on C, that
a morphism g: B -> C in the functor category C^I has a
c^I-image, where c^I
is the pre-cover relation on C^I induced by c, as soon as each
component of g has a c-image.
We then apply this to show that if a pointed category C is: (i)
algebraically cartesian closed; (ii) exact protomodular and action accessible; or
(iii) admits normalizers, then the same is true of each functor category C^I
with I finite. In addition, our results give explicit constructions of
c^I-images in
functor categories using limits and c-images in the underlying category.
In particular, they can be used to give explicit constructions of
both centralizers and normalizers in functor categories using limits and
centralizers or normalizers (respectively) in the underlying category.

Keywords:
cover relation, image of cover relation, semi-abelian category, normalizer, centralizer, action accessible, functor category

2020 MSC:
18E13, 18A25, 18D99

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 10, pp 311-318.

Published 2022-02-02.

http://www.tac.mta.ca/tac/volumes/38/10/38-10.pdf

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