In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the co duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets. Now, let D 2 be the suspension of D 1 . This is the ω-category generated by the following 2-graph: 0 Denis-Charles Cisinski and Carlos Simpson for helpful discussions during the development of this project. Finally, we thank Marnie Valentini for her help with English. WarningThis text is still in draft form. Although the author is convinced of the veracity of the results, the text probably still contains many typos, and some proof are not written in the clearest way, especially the more technical ones. A new version should quickly replace this one.

CONDITIONS DE KAN SUR LES NERFS DES ω-CAT ÉGORIESF ÉLIX LOUBATON Résumé. On montre que le nerf de Street d'une ω-catégorie stricte C est un complexe de Kan (respectivement une quasi-catégorie) si et seulement si les n-cellules de C pour n ≥ 1 (respectivement n > 1) sont faiblement inversibles. De plus, on définit une structure d'ensemble complicial sur le nerf de C. L'ensemble complicial N (C) est alors n-trivial si et seulement si les k-cellules de C pour k ≥ n sont faiblement inversibles.

We construct a Quillen equivalence between marked simplicial sets and (marked) Segal precategories enriched in marked simplicial sets. From this, we can deduce that Verity's n-complicial sets are a model of (∞, n)-categories in the sense of Barwick and Schommer-Pries.