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Discrete double fibrations. (English) Zbl 1464.18020

Ordinary fibrations have their analogues in \(2\)-fibrations over a fixed base \(2\)-category [M. Buckley, J. Pure Appl. Algebra 218, No. 6, 1034–1074 (2014; Zbl 1296.18006)]. Discrete fibrations over a fixed base \(2\)-category are of their \(2\)-dimensional version in discrete \(2\)-fibrations [M. Lambert, “Discrete 2-fibrations”, Preprint, arXiv:2001.11477]. They are justified by the existence of a representation theorem taking the form of an equivalence with representations of the base structure via a category of elements construction.
Within the double-categorical realm, R. Paré [Theory Appl. Categ. 25, 436–489 (2011; Zbl 1251.18004)] proposed that certain span-valued, lax functors on a double category \(\mathbb{B}\)are presheaves on \(\mathbb{B}\). The principal objective in this paper is to isolate the notion of discrete double fibration corresponding to this notion of presheaf. Paré’s double category of elements is exploited to exhibit a representation theorem as above. What is technically interesting is that an equivalence of virtual double categories is ultimately achieved, for which the language of monoids and modules in virtual double categories [T. Leinster, Higher operads, higher categories. Cambridge: Cambridge University Press (2004; Zbl 1160.18001)] is in use.

MSC:

18N10 2-categories, bicategories, double categories
18N25 Categorification
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References:

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[71] Ross Street, Macquarie University:ross.street@mq.edu.au
[72] Tim Van der Linden, Universit´e catholique de Louvain:tim.vanderlinden@uclouvain.be
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