A cubical model for $(\infty, n)$-categories

Abstract

We propose a new model for the theory of (∞, n)-categories (including the case n = ∞) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our model is shown to be monoidal with respect to suitably defined (lax and pseudo) Gray tensor products; in particular, these tensor products are both associative and biclosed. Furthermore, we show that the triangulation functor to pre-complicial sets is a left Quillen functor and is strong monoidal with respect to both Gray tensor products.

Equivalence of cubical and simplicial approaches to $(\infty,n)$-categories

“…Lastly, we may extend the triangulation functor T ∶ cSet → sSet to the marked setting, following [CKM20]. To do this, we first need an explicit description of the simplices of T ◻ n = (∆ 1 ) n = N [1] n .…”

“…Then in Section 2, we introduced our comical sets, modifying the definition of Date: June 18, 2021. [CKM20]. We show that the marked triangulation functor T + ∶ cSet + → sSet + is a left Quillen functor in Section 3.…”

“…D]), using marked simplicial sets, i.e., simplicial sets in which certain simplices are formally designated as equivalences, witnessing composition of lower-dimensional simplices. The second model,in comical sets, introduced in [CKM20], is based on marked cubical sets instead and is intended to provide an alternative to complicial sets, particularly well suited for the treatment of Gray tensor product.…”

“…Both of these models arise as fibrant-cofibrant objects in certain model structures on the categories sSet + of marked simplicial sets and cSet + of marked cubical sets, respectively. In [CKM20], a triangulation functor T ∶ cSet + → sSet + is constructed as an extension of the usual triangulation functor T ∶ cSet → sSet [Cis06] to the marked setting and it is conjectured that this functor is a Quillen equivalence.…”

“…We prove a slight modification of this conjecture. In our proof, we change the definition of a comical set from the one given in [CKM20] by adjusting one of the pseudo-generating maps, to require that the markings on the standard (i, ε)-cube ◻ n i,ε be more restrictive. As in [CKM20], we construct a model structure whose fibrant-cofibrant objects are the comical sets.…”