nLab multisimplicial set

Idea

Multisimplicial sets are the analogs of simplicial sets with simplices replaced by multisimplices? (perhaps more appropriately called multiplexes).

Definition

Definition

An $n$-fold multisimplicial set is a presheaf on the $n$-fold multisimplex category? $\Delta^n$, that is, a functor $X\colon(\Delta^n)^{op}\to Sets$, equivalently a multisimplicial object? in the category Set of sets.

Relation to simplicial sets

The category of $n$-fold multisimplicial sets can be equipped with a model structure that turns it into a model category that is Quillen equivalent to the standard Kan–Quillen model structure on simplicial sets.

Cofibrations of multisimplicial sets are precisely monomorphisms. Weak equivalences are induced from simplicial sets by the diagonal functor.

The corresponding Quillen adjunction is constructed as as a nerve-realization adjunction? for the functor

$$\Delta^n\to sSet$$

that sends a multisimplex $(m_1,\ldots,m_n)$ to $\Delta^{m_1}\times\cdots\times\Delta^{m_n}$.

The left adjoint is given by a left Kan extension. The right adjoint sends a simplicial set $X$ to its multisimplicial nerve, which sends a multisimplex $(m_1,\ldots,m_n)$ to $sSet(\Delta^{m_1}\times\cdots\times\Delta^{m_n},X)$.

To see that this Quillen adjunction is a Quillen equivalence, one can either argue directly, by computing the derived unit and derived counit and showing that they are weak equivalences, or, much more elegantly, invoke a theorem by Grothendieck and Maltsiniotis.

The latter theorem (Maltsiniotis, Proposition 1.6.8) states that a totally aspherical small category $C$ with a separating aspherical interval is a strict test category, and, therefore, the category of presheaves of sets on $C$ admits a model structure whose cofibrations are monomorphisms and weak equivalences are induced by the category of elements functor from the Thomason model structure on the category of small categories. Furthermore, the resulting model category of presheaves of sets on $C$ is a cartesian combinatorial model category that is Quillen equivalent to the Kan–Quillen model structure on the category of simplicial sets.

We now verify the conditions of the theorem:

• The category of multisimplices is nonempty.

• The categorical product of two representable multisimplicial sets is acyclic, i.e., weakly equivalent to the terminal object.

• The multisimplex $([1],[1],\ldots,[1])$ together with the two canonical inclusions of $([0],\ldots,[0])$ is a separating aspherical interval.

Exterior product

An important operation on multisimplicial sets is the exterior product

$$Fun((\Delta^m)^{op},Set)\times Fun((\Delta^n)^{op},Set)\to Fun((\Delta^{m+n})^{op},Set)$$

defined as the left Kan extension of the tautological functor

$$\Delta^m\times\Delta^n\to\Delta^{m+n}.$$

The exterior product is a left Quillen bifunctor whose left derived bifunctor? model the cartesian product in the ∞-category of spaces.

The exterior product is useful when it is desirable to have a product operation that does not require subdivision.

Related concepts

• simplicial sets

• cubical sets

• Cisinski model category

• test category, strict test category

References

• Zouhair Tamsamani?, On non-strict notions of $n$-category and $n$-groupoid via multisimplicial sets (arXiv:alg-geom/9512006)

• Gerd Laures? and James E. McClure,

  Multiplicative properties of Quinn spectra (arXiv:0907.2367v2)

• Yifeng Liu and Weizhe Zheng, Gluing restricted nerves of $\infty$-categories (arXiv:1211.5294)

• Georges Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque 301 (2005).