Cubical models of $\infty$-presheaves and the Bousfield-Kan formula
Kensuke Arakawa, Daniel Carranza, Chris Kapulkin
TL;DR
This work develops covariant and cocartesian model structures on slice categories of cubical sets and marked cubical sets, extending the Grothendieck-style fibration formalism to the cubical setting. It establishes invariance under base weak equivalences, provides explicit cubical straightening–unstraightening results over nerves of ordinary categories, and proves a cubical Bousfield–Kan formula for homotopy colimits in monoidal model categories satisfying Muro’s unit axiom. The results connect cubical and simplicial higher-categorical formalisms via triangulation, while delivering descent, base-change, and comparison theorems essential for practical computations in cubical models of ∞-categories. Collectively, they offer a robust cubical framework for ∞-presheaves and Grothendieck constructions with broad applicability to higher category theory and homotopy theory.
Abstract
We construct the covariant and the cocartesian model structures on the slice categories of cubical sets and marked cubical sets, respectively. As an application, we derive a version of the Bousfield-Kan formula for arbitrary cofibrantly generated monoidal model categories satisfying Muro's axiom.