Added a new section:

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 (\cite{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.

So one can work around the bug via subtle changes in whitespace. Hardly a permanent solution, though. Is Instiki still being maintained? How does one report a bug in it?

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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 nLab/nForum likes double dollar expressions to be on their own line, giving the following. (I’ve updated the nLab page)

]]>An important operation on multisimplicial sets is the

$Fun((\Delta^m)^{op},Set)\times Fun((\Delta^n)^{op},Set)\to Fun((\Delta^{m+n})^{op},Set)$exterior productdefined as the left Kan extension of the tautological functor

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

I created the article multisimplicial set.

It seems that I triggered some bug in Instiki that prevents it from rendering displayed formulas correctly.

The article compiles just fine in TeX, so this is clearly a bug.

In fact, Instiki seems to insert some new text:

definedasthechunk29161400wikichunklinkchunkofthetautologicalfunctor

All this “chunk” and “wikichunklink” stuff is clearly a bug.

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