A double $(\infty ,1)$-categorical nerve for double categories
[Un nerf $(\infty ,1)$-catégorique double pour les catégories doubles]
Moser, Lyne1
1Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Annales de l'Institut Fourier, Online first, 81 p.

We construct a nerve from double categories into double $(\infty ,1)$-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories and the model structure on bisimplicial spaces for double $(\infty ,1)$-categories seen as double Segal objects in spaces complete in the horizontal direction. We then restrict the nerve along a homotopical horizontal embedding of $2$-categories into double categories, and show that it gives a right Quillen and homotopically fully faithful functor between Lack’s model structure for $2$-categories and the model structure for $2$-fold complete Segal spaces. We further show that Lack’s model structure is right-induced along this nerve from the model structure for $2$-fold complete Segal spaces.

On construit un nerf des catégories doubles dans les $(\infty ,1)$-catégories doubles et prouve que cela réalise un foncteur de Quillen à droite qui est homotopiquement pleinement fidèle entre la catégorie de modèles pour les catégories doubles faiblement horizontalement invariantes et la catégorie de modèles sur les espaces bisimpliciaux pour les $(\infty ,1)$-catégories doubles vues comme des espaces de Segal doubles qui sont complets dans la direction horizontale. On restreint ensuite ce nerf le long d’un plongement horizontal homotopique des $2$-catégories dans les catégories doubles et prouve que cela réalise un foncteur de Quillen à droite qui est homotopiquement pleinement fidèle entre la catégorie de modèles de Lack sur les $2$-catégories et la catégorie de modèles pour les espaces de Segal complets doubles. On montre de plus que la catégorie de modèles de Lack sur les $2$-catégories peut être obtenue comme la catégorie de modèles transferrée le long de ce nerf depuis la catégorie de modèles pour les espaces de Segal complets doubles.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3750
Classification : 18N10, 18N25, 18N65, 55U35
Keywords: Nerve, 2-categories, double categories, 2-fold complete Segal spaces, double $\infty $-categories
Mots-clés : Nerf, 2-catégories, catégories doubles, espaces de Ségal complets doubles, $\infty $-catégories doubles

Moser, Lyne  1

1 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany 
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Moser, Lyne. A double $(\infty ,1)$-categorical nerve for double categories. Annales de l'Institut Fourier, Online first, 81 p.

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