Symmetric Cubical Sets

TL;DR

This paper introduces a new cubical model for homotopy types using a category Qs that extends the classical box category, enabling better modeling of homotopy categories and enriched structures.

Contribution

It defines the category Qs as a PROP containing the classical box category, providing a new framework for modeling homotopy types with symmetric monoidal structures.

Findings

Qs-Set models the homotopy category effectively.

Enrichments in Qs-Set are homotopically well-behaved.

The framework generalizes classical cubical models.

Abstract

We introduce a new cubical model for homotopy types. More precisely, we'll define a category Qs with the following features: Qs is a PROP containing the classical box category as a subcategory, the category Qs-Set of presheaves of sets on Qs models the homotopy category, and combinatorial symmetric monoidal model categories with cofibrant unit all have homotopically well behaved Qs-Set enrichments.