Geodesic
On spans with right fibred right adjoints
Theory and applications of categories, Tome 34 (2019), pp. 854-882

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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.

Publié le :
Classification : 18A05, 18A22, 18A25, 18A40, 18A99, 18B99, 18D99
Keywords: action representable, locally algebraically cartesian closed, semi-abelian, split extension, normalizer, prefibration, right fibred right adjoints, regular span 
@article{TAC_2019_34_a27,
     author = {J. R. A. Gray},
     title = {On spans with right fibred right adjoints},
     journal = {Theory and applications of categories},
     pages = {854--882},
     publisher = {mathdoc},
     volume = {34},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a27/}
}
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J. R. A. Gray. On spans with right fibred right adjoints. Theory and applications of categories, Tome 34 (2019), pp. 854-882. http://geodesic.mathdoc.fr/item/TAC_2019_34_a27/