A cubical model for (∞,n)-categories

Campion, Tim¹ ; Kapulkin, Krzysztof² ; Maehara, Yuki³

¹Department of Mathematics, Johns Hopkins University, Baltimore, MD, United States
²Department of Mathematics, University of Western Ontario, London, ON, Canada
³Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

Geometry & topology, Tome 29 (2025) no. 3, pp. 1115-1170

Résumé

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 precomplicial sets is a left Quillen functor and is strong monoidal with respect to both Gray tensor products.

Détail

DOI : 10.2140/gt.2025.29.1115

Keywords: cubical set, $(\infty,n)$-category, Gray tensor product, model category, triangulation

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