Dualities in the complicial model of $\infty$-categories

Abstract

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.

$n$-Complicial sets as a model of $(\infty,n)$-categories

“…By adjoint property, this implies that f 1 : C 1 → D 1 has the right lifting property against ∂G co n → G co n and G n → t n (G n ). According to proposition 3.2.5 of [Lou22], f 1 is then a trivial fibration, and so a weak equivalence. The morphism f is then a equivalence of marked Segal A-categories.…”

“…These results, by establishing links between some (∞, ω)-categories and their associated strict ω-categories, allow to solve complex coherence problems. In particular, proposition 4.1.3 of [Lou22] which gives a characterization of some self-equivalence of marked simplicial sets and which is used in the proof of theorem 0.0.2, depends strongly on them.…”

A Quillen adjunction between globular and complicial approaches to (∞,n)-categories

“…By Definition 2.2.5, saying that F is an (∞, n)-equivalence amounts to being a hom-wise equivalence of (∞, n − 1)-categories and essentially surjective up to (∞, n − 1)-equivalence. By [Lou22a,Corollary 3.2.11] this is equivalent to saying that F is a weak equivalence in msSet (∞,n) . Using [Hov99, Proposition 1.2.8], this is equivalent to saying that the map F ∶ A → B is a homotopy equivalence in msSet (∞,n) , meaning that there exist a map G∶ B → A and homotopies in msSet (∞,n)…”

What is an equivalence in a higher category?

“…By Definition 2.2.5, saying that 𝐹 is an (∞, 𝑛)-equivalence amounts to being a hom-wise equivalence of (∞, 𝑛 − 1)-categories and essentially surjective up to (∞, 𝑛 − 1)equivalence. By [77,Corollary 3.2.11] this is equivalent to saying that 𝐹 is a weak equivalence in 𝑚𝑠𝑒𝑡 (∞,𝑛) . Using [55, Proposition 1.2.8], this is equivalent to saying that the map 𝐹 ∶ 𝒜 → ℬ is a homotopy equivalence in 𝑚𝑠𝑒𝑡 (∞,𝑛) , meaning that there exist a map 𝐺 ∶ ℬ → 𝒜 and homotopies in 𝑚𝑠𝑒𝑡 (∞,𝑛)…”