Horn of a Simplicial Set

Question

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:

1. 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.

2. 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?

Answer 1

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".