Given a simplicial set $X\_\bullet$, define its **powerset simplicial set** $\mathcal{P}\_\bullet(X)$ as the composition
$$\Delta^\mathsf{op}\xrightarrow{X\_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\mathsf{Set},$$
where $\mathcal{P}$ is the covariant powerset functor.


**How homotopically well-behaved is the powerset simplicial set construction?**


1. If $X\_\bullet$ is a Kan complex, must $\mathcal{P}\_\bullet(X)$ also be one?
2. Given a Kan complex $X\_\bullet$, are the homotopy groups of $X\_\bullet$ related to those of $\mathrm{Ex}^\infty\_\bullet(\mathcal{P}\_\bullet(X))$?



Homotopical properties of powersets of simplicial sets
The first question has a negative answer, given by the simplicial set $\def\Exi{{\sf Ex}^{\sf\infty}}X=\Exi Y$, where $Y$ is a simplicial set generated by vertices $a,b,b',c,c',d,d'$, 1-simplices $ab,ab',ac,ac',ad,ad'$, 2-simplices $abc,acd,abd,ab'c',ac'd',ab'd',abc'$, and 3-simplices $abcd,ab'c'd'$.


We specify a 3-horn $Λ^3\_0→P(X)$ by setting its 1st face to $\{acd,ac'd'\}$, 2nd face to $\{abd,ab'd'\}$, 3rd face to $\{abc,ab'c',abc'\}$.


Any 3-simplex of $P(X)$ that fills in this horn must in particular contain a 3-simplex of $X$ whose 3rd face is $abc'$. This forces its 1st face to be $ac'd'$ and its 2nd face to be $abd$, which means that the 1-simplex connecting 0th and 3rd vertices would have to be simulataneously $ad$ and $ad'$, which is impossible.


