My question refers to description of horns $\Lambda^n_k$ for Kan Fibrations in Laures' and Szymik's "Grundkurs Topologie" (page 227). Sorry, there exist only a German version. Here the relevant excerpt:
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DESCRIPTION (X): Namely for $0 \le r \le n$ the $r$-th horn $\Lambda^n_k$ is defined as simplicial subset of $\Delta^n = Hom_{\Delta}(-, [n])$ which has as $m$-simplices exactly the order preserving maps $f:[m] \to [n]$ having $r$ not in the image.
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DESCRIPTION (XX): Geometrically (so in sense of a gemetrical realization), according to the author this horn coinsides with $n$-standard simplex $\Delta^n_{top}= \vert \Delta^n \vert $ after removing to inner points and the face lying on the opposite to the point $r$.
In the excerpt on Abb 11.3 (left) there is an image of such horn for $n=2$ and $r=0$.
What I don't understand is why this geometric description coinsides exactly with the first one (X)
Here I see following problem: the non degenerated $1$-simplces of $\Delta^2(1) = Hom_{\Delta}([1], [2])$ of $\Delta^2$ are exactly the three face maps $d^i[1] \to [2]$ for $i=0,1,2$ defined via
$$
d^i(m) =
\begin{cases}
m, & \text{if }m < i \\
m+1, & \text{if }m \ge i
\end{cases}
$$
But in this case the only (non degenerated) $1$-simplex of $\Delta^2$ not containing $r=0$ in the image is $d^0$ by mapping $0 \mapsto 1, 1 \mapsto 2$.
After realization the non degenerated $1$-simplices of $\Delta^2$ are exactly the $1$-faces of the triangle in the picture where the $d^i$ correspond to the arrows from $d^i(0)$ to $d^i(1)$.
Using this correspondence the realization of the horn $\Lambda^2 _0$ should have as a $1$-face only the arrow from $1$ to $2$ (corresponding to $d^0$).
But in the image which correspond to description from (XX) the subset $\vert \Lambda^2 _0 \vert$ of the tringle $\Delta^2_{top}= \vert \Delta^2 \vert $ has the arrows $0 \to 1$ and $0 \to 2$ as faces which correspond by the realization exactly to $d^1$ and $d^2$. But these have $r=0$ as image.
So I think that the descriptions (X) and (XX) of the horn cannot coinside, right?
Or is there an error in my reasonings?
Best Answer
This community wiki solution is intended to clear the question from the unanswered queue.
As Arnaud D. remarked in his comment, the definition of a horn in the book by Laures and Szymik is false. In fact, the $r$-th horn is the simplicial subset of $\Delta^n$ which has as $m$-simplices exactly the order preserving maps $f:[m] \to [n]$ having $r$ in the image.
Geometrically the $r$-th horn is obtained from $\Delta^n$ by removing two simplices, namely the only $n$-simplex $id : [n] \to [n]$ and the unique $(n-1)$-simplex $s_r : [n-1] \to [n]$ with $r \notin s_r([n-1])$ (the face opposite to $r$). For obviuus reasons some German authors also use the word "Trichter" instead of "Horn".