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.