A Thomason Model Structure on the Category of Small n-fold Categories

We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. This is an n-fold analogue to Thomason’s Quillen model structure on Cat. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.

💡 Research Summary

The paper establishes a cofibrantly generated Quillen model structure on the category of small n‑fold categories (denoted n‑Cat) and proves that this structure is Quillen equivalent to the classical Kan–Quillen model structure on simplicial sets (SSet). The construction is a direct higher‑dimensional analogue of Thomason’s model structure on the ordinary category Cat.

The authors begin by defining the n‑fold nerve functor
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