nLab cubical-type model category

General

These exist a class of model category structures on certain categories of cubical sets, due to Christian Sattler, motivated by cubical type theory.

Assumptions

Let $\mathbb{C}$ be a small category with finite products. Assume there is a (cartesian) interval object $\mathbb{I}$, i.e., an object $\mathbb{I}$ with two parallel morphisms $0,1 \colon e \to \mathbb{I}$, where $e$ is the terminal object in $\mathbb{C}$.

Assume further that we have connection maps $\vee,\wedge : \mathbb{I}\times\mathbb{I} \to \mathbb{I}$ such that $x \vee 0 = 0 \vee x = x$ and $x \wedge 1 = 1 \wedge x = x$.

We also need a face lattice $\mathbb{F}$, that is, a sub-lattice of the subobject classifier $\Omega$ in the presheaf category $cSet = \mathbb{C}^{op} \to Set$ containing the endpoints of each cylinder object $J^+ = J \times \mathbb{I}$ for each object $J$ of $\mathbb{C}$, and an operation $\forall \colon \mathbb{F}^\mathbb{I} \to \mathbb{F}$ right adjoint to the projection in the sense that $\psi \le \forall \delta$ iff $\psi p \le \delta$, where $p : \mathbb{F} \times \mathbb{I} \to \mathbb{F}$.

If we want the model to be effective, we also require that each proposition $\psi = 1$ with $\psi \in \mathbb{F}(I)$ is decidable. (This rules out taking $\mathbb{F} = \Omega$.)

Instances of the assumptions

A simple and central example is the full subcategory of the category of posets $Pos$ on powers of $\mathbb{I} = (0 \lt 1)$. This is also the Lawvere theory of distributive lattices. The idempotent completion is the full subcategory of $Pos$ on finite lattices.

Other prominent examples are the Lawvere theories of de Morgan, Kleene, and boolean algebras.

For computational purposes we can take $\mathbb{F}$ to be the smallest lattice containing the cylinder endpoints (the faces of cubes). To get Cisinski model structures we can take $\mathbb{F} = \Omega$.

Equivalence with simplicial sets

One may wonder whether these models structures are equivalent to the model in simplicial sets. This is not the case for Cartesian cubes; see mailing-list. However, this model does form a Grothendieck (infinity,1)-topos, because it satisfies the Giraud axioms. For deMorgan cubical sets the geometric realization is not a Quillen equivalence (because of the reversal map); the counterexample is the unit interval quotient by the symmetry); see Sattler. Whether they are equivalent by another map is not yet excluded. The question whether the geometric realization for the cubical sets based on distributive lattices is an equivalence is still open, cf. also Hackney and Rovelli and Streicher and Weinberger.

References

• Thierry Coquand, A model structure on some presheaf categories, pdf

• Thierry Coquand, Some examples of complete Cisinski model structures, pdf

• Christian Sattler, Do cubical models of type theory also model homotopy types, lecture at Hausdorff Trimester Program: Types, Sets and Constructions, youtube

• Philip Hackney and Martina Rovelli, Induced model structures for ∞-categories and ∞-groupoids, $[$arXiv:2102.01104$]$, Proc. Amer. Math. Soc., June 10, 2022

• Thomas Streicher and Jonathan Weinberger, Simplicial sets inside cubical sets $[$arXiv:1911.09594$]$, Theory and Applications of Categories, Vol. 37, 2021, No. 10, pp 276–286