[Paper Review] A co-reflection of cubical sets into simplicial sets with applications to model structures

Chris Kapulkin, Zachery Lindsey+1 more|arXiv (Cornell University)|Jun 21, 2019

Homotopy and Cohomology in Algebraic Topology8 references4 citations

TL;DR

This paper establishes a co-reflection of simplicial sets into cubical sets with connections via a straightening functor, enabling the transfer of cofibrantly generated model structures—such as Quillen’s and Joyal’s—onto cubical sets. The key contribution is a new model structure on cubical sets that presents the homotopy theory of $(\infty,1)$-categories, providing the first such model in the cubical setting.

ABSTRACT

We show that the category of simplicial sets is a co-reflective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reflector, one can transfer any cofibrantly generated model structure in which cofibrations are monomorphisms to cubical sets, thus obtaining cubical analogues of the Quillen and Joyal model structures.

Motivation & Objective

• To establish a co-reflection of simplicial sets into cubical sets with connections, enabling model structure transfers.
• To provide a cubical analogue of the Quillen and Joyal model structures on simplicial sets.
• To construct a model of the homotopy theory of $(\infty,1)$-categories in the category of cubical sets.
• To show that the Grothendieck construction (unstraightening) serves as a co-reflector for the inclusion of simplicial sets into cubical sets.
• To prove that the induced model structures on cubical sets are Quillen equivalent to their simplicial counterparts.

Proposed method

• The authors define a straightening functor $\mathscr{Q}$ from simplicial sets to cubical sets with connections, which serves as a fully faithful left adjoint to the unstraightening functor $\int$.
• They prove that $\mathscr{Q}$ exhibits simplicial sets as a co-reflective subcategory of cubical sets with connections, using the Grothendieck construction as the co-reflector.
• Using the right-induced model structure construction from [HKRS17, GKR18], they transfer cofibrantly generated model structures from simplicial sets to cubical sets when cofibrations are monomorphisms.
• The transfer relies on verifying that the unit of the adjunction $\mathscr{Q} \dashv \int$ is a weak equivalence, which is established via Lemma 3.8.
• They compare the resulting model structures on cubical sets with the existing Grothendieck model structure for $\infty$-groupoids, showing Quillen equivalence via geometric realization and simplicial homotopy types.
• The proof leverages the fact that $\mathrm{T} \mathscr{Q} \Lambda^n_i$ and $\mathrm{T} Q^n$ are contractible simplicial sets, with $|\mathrm{T} Q^n| \simeq \Delta^n$ and $|\mathrm{T} \mathscr{Q} \Lambda^n_i| \simeq |\Lambda^n_i|$.

Experimental results

Research questions

• RQ1 Can the category of simplicial sets be embedded as a co-reflective subcategory of cubical sets with connections?
• RQ2 Does the straightening functor induce a co-reflection that allows transfer of model structures from simplicial to cubical sets?
• RQ3 Can a cubical model structure for $(\infty,1)$-categories be constructed via this co-reflection?
• RQ4 Is the resulting model structure on cubical sets Quillen equivalent to the Joyal model structure on simplicial sets?
• RQ5 How does the new cubical model structure compare to the Grothendieck model structure for $\infty$-groupoids?

Key findings

• The straightening functor $\mathscr{Q}$ provides a fully faithful embedding of simplicial sets into cubical sets with connections, making simplicial sets a co-reflective subcategory.
• The unstraightening functor $\int$ serves as the co-reflector, and the unit of the adjunction $\mathscr{Q} \dashv \int$ is a weak equivalence, as shown in Lemma 3.8.
• The right-induced model structure on cubical sets is Quillen equivalent to the original model structure on simplicial sets, provided cofibrations are monomorphisms.
• The resulting model structure $\mathsf{cSet}_{IJ}$ on cubical sets is the first known model for the homotopy theory of $(\infty,1)$-categories in the cubical setting.
• The identity functor forms a Quillen equivalence between $\mathsf{cSet}_{IQ}$ (the right-induced Quillen model) and $\mathsf{cSet}_G$ (the Grothendieck model for $\infty$-groupoids).
• Geometric realization shows that $|\mathrm{T} Q^n| \simeq \Delta^n$ and $|\mathrm{T} \mathscr{Q} \Lambda^n_i| \simeq |\Lambda^n_i|$, confirming that the induced fibrations are weak equivalences.