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According to this question, there is a model structure on $\mathrm{Set}$ in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are the maps f:X→Y such that X and Y are both empty or both nonempty.

As $\Delta^{op}$ is a Reedy category. We can consider the model structure induced on $\mathrm{Set}^{\Delta^{op}}$. How does this model structure compare to the well known model structures of $\mathrm{sSet}$ (Quillen, Joyal)?

There is probably little hope, but can we hope a structure on $\mathrm{Set}$ which induces the Quillen model structure?

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The weak equivalences in the Reedy model structure are levelwise, and in a simplicial set if there is a simplex of any dimension then there are simplices of all dimensions. Thus, the weak equivalences are the simplicial maps $X\to Y$ in which $X$ and $Y$ are either both empty or both nonempty; so the homotopy theory is not very interesting.

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There are only nine possible model structures on $\mathbf{Set}$. You can easily check that none of them give the right weak equivalences:

  • Either everything is a weak equivalence,
  • or $X \to Y$ is a weak equivalence if and only if both $X$ and $Y$ are empty or both $X$ and $Y$ are non-empty,
  • or only isomorphisms are weak equivalences.
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