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 
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/
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     author = {J. R. A. Gray},
     title = {On spans with right fibred right adjoints},
     journal = {Theory and applications of categories},
     pages = {854--882},
     year = {2019},
     volume = {34},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a27/}
}
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