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On spans with right fibred right adjoints

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J. R. A. Gray

We introduce a new condition on an abstract span of categories which
we refer to as having right fibred right adjoints, RFRA for short.
We show that:

(a) the *span of split extensions* of a semi-abelian category
C has RFRA if and only if C is action representable;

(b) the *reversed span* to the one considered in (a) has
RFRA if and only if C is locally algebraically cartesian closed;

(c) the span of split extensions of the category of morphisms of C
has RFRA if and only if C is action representable and has
normalizers;

(d) the reversed span to the one considered in (c) has
RFRA if and only if C is locally algebraically cartesian closed.

We also examine our condition for the span of monoid actions (of
monoids in a monoidal category C on objects in a given category on
which C acts), and for various other spans.

Keywords:
action representable, locally algebraically cartesian closed,
semi-abelian, split extension, normalizer, prefibration, right fibred
right adjoints, regular span

2010 MSC:
18A05, 18A22, 18A25, 18A40, 18A99, 18B99, 18D99

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 28, pp 854-882.

Published 2019-09-20.

http://www.tac.mta.ca/tac/volumes/34/28/34-28.pdf

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