In this paper on Page 21, the first line of the proof, Peter Scholze seems to claim that any hypercover, consisting of finite sets, splits. I find this hard to believe.

I am not familiar with categorical topology, but let's consider the constant simplicial set, where you map all simplices to a point and all arrows to the identity. It seems to me that this is a counterexample, is it not? And if not, why is Scholze's claim true?

EDIT: I need to explain my question a bit: According to this source, a simplicial object $U_\bullet$ splits if for every $n$ there exists a subobject $NU_n$ of $U_n$ such that the map $$ \coprod_{\phi:[n]\twoheadrightarrow [m]}NU_n\to U_n $$ is an isomorphism. In my example we have $U_n=*$ the one pointed set. It has exactly two subobjects, the empty set or the point. The subobject cannot be the empty set, because then the coproduct (=disjoint union) would be empty, too. So it has to be $*$. In wich case we get an isomorphism between the coproduct of several copies of $*$ and $*$, which is absurd.

In this paper on Page 21, the first line of the proof, Peter Scholze seems to claim that any hypercover, consisting of finite sets, splits. I find this hard to believe.

I am not familiar with categorical topology, but let's consider the constant simplicial set, where you map all simplices to a point and all arrows to the identity. It seems to me that this is a counterexample, is it not? And if not, why is Scholze's claim true?

EDIT: I need to explain my question a bit: According to this source, a simplicial object $U_\bullet$ splits if for every $n$ there exists a subobject $NU_n$ of $U_n$ such that the map $$ \coprod_{\phi:[n]\twoheadrightarrow [m]}NU_m\to U_n $$ is an isomorphism. In my example we have $U_n=*$ the one pointed set. It has exactly two subobjects, the empty set or the point. The subobject cannot be the empty set, because then the coproduct (=disjoint union) would be empty, too. So it has to be $*$. In wich case we get an isomorphism between the coproduct of several copies of $*$ and $*$, which is absurd.

In this paper on Page 21, the first line of the proof, Peter Scholze seems to claim that any hypercover, consisting of finite sets, splits. I find this hard to believe.

I am not familiar with categorical topology, but let's consider the constant simplicial set, where you map all simplices to a point and all arrows to the identity. It seems to me that this is a counterexample, is it not? And if not, why is Scholze's claim true?

In this paper on Page 21, the first line of the proof, Peter Scholze seems to claim that any hypercover, consisting of finite sets, splits. I find this hard to believe.

I am not familiar with categorical topology, but let's consider the constant simplicial set, where you map all simplices to a point and all arrows to the identity. It seems to me that this is a counterexample, is it not? And if not, why is Scholze's claim true?

EDIT: I need to explain my question a bit: According to this source, a simplicial object $U_\bullet$ splits if for every $n$ there exists a subobject $NU_n$ of $U_n$ such that the map $$ \coprod_{\phi:[n]\twoheadrightarrow [m]}NU_n\to U_n $$ is an isomorphism. In my example we have $U_n=*$ the one pointed set. It has exactly two subobjects, the empty set or the point. The subobject cannot be the empty set, because then the coproduct (=disjoint union) would be empty, too. So it has to be $*$. In wich case we get an isomorphism between the coproduct of several copies of $*$ and $*$, which is absurd.