ScalingStacks

Because the usual examples of closed model categories are not small categories, their classification diagrams are not bisimplicial sets. We may elude this difficulty by positing, after Grothendieck, the existence of a universe $U$ (a model for set theory) in which ${\operatorname{\mathbf{M}}}$ is defined. Then $N({\operatorname{\mathbf{M}}},{\operatorname{\mathbf{W}}})$ is an honest simplicial space (though not modeled in the universe $U$, but rather in some higher universe $U^{\prime}$).

Alternately, we note that there is no difficulty if the model category ${\operatorname{\mathbf{M}}}$ is a small category, and that such exist in practise. As an example, choose an uncountable cardinal $\gamma$, and let ${\operatorname{\mathcal{S}}}_{\gamma}$ denote a skeleton of the category of all simplicial sets which have fewer than $\gamma$ simplices. Then ${\operatorname{\mathcal{S}}}_{\gamma}$ is a small category, and is in fact a simplicial closed model category. (Of course, the category ${\operatorname{\mathcal{S}}}_{\gamma}$ is not suitable for all purposes; for example, it is not cartesian closed.)