Concretely, what is the horn $\Lambda_k^n$?

Question

Recall that we define the simplex category $\mathbf{\Delta}$ and define the category of simplicial sets $\mathbf{\Delta\mbox{-}Set}$ to have as objects, functors $\mathbf{\Delta}^\mathrm{op}\to\mathbf{Set}$, and as morphisms, natural transformations between these functors. We then define, for $n\in\mathbb{N}$, the simplicial n-simplex $\Delta^n$ s.t. $\Delta^n(i)=\mathrm{Hom}_\mathbf{\Delta}([i],[n])$ with the induced face and degeneracy maps.

Then we define the $(n,k)$-horn to be the minimal simplicial subset $\Lambda_k^n$ of $\Delta^n$ such that $$\Big(\delta^i:[n-1]\to[n]\Big)\in\Lambda_k^n(n-1)$$ for all $i\neq k$.

However, I'm having trouble making any concrete statements about $\Lambda_k^n$. How do we know that $\Lambda_k^n\ne\Delta^n$? How can it be shown that the non-degenerate simplices of $\Lambda_k^n$ do in fact form a horn, as the name suggests?

Answer 1

A more hands-on method of seeing that the horns aren't the whole simplicial set is to observe that $\Lambda^n_k\subset \Delta^n$ is generated by $n-1$-simplices, in the sense that all of its $m$-simplices for $m\geq n$ are degenerate. This is clear from the definition, and it seems hopefully clear that this condition is invariant under isomorphism (simplicial maps preserve degenerate simplices.) But of course $\Delta^n$ has a rather notable nondegenerate $n$-simplex, the identity map! Note that this isn't degenerate, since it doesn't factor through $n-1$.

Answer 2

The sharpest description I found is the one given at Exercise 1.1.2.14 of Kerodon (it can be found elsewhere).

A horn $\Lambda^n_k\to X$ is a sequence $(\sigma_0,\dots,\sigma_{k-1},\bullet,\sigma_{k+1},\dots,\sigma_n)$ of $(n-1)$-simplices such that $d_i(\sigma_j) = d_{j-1}(\sigma_i)$ whenever $i<j$ and $i\neq k\neq j$.

When you picture $\Lambda^n_k$ as the $n$-simplex $\Delta^n$ with its $k$-th face missing, the $\sigma_i$ represent the faces (notice the $k$-th input is missing), and $d_i(\sigma_j) = d_{j-1}(\sigma_i)$ is saying that, as expected, "the $k$-th face of $\sigma_j$ is the $(j-1)$-face of $\sigma_i$".

For instance, a horn $\Lambda^2_0\to X$ is a sequence $(\bullet,f,h)$, where $f,h\in X_1$ are such that $d_1(f) = d_1(h)$. This horn (in fact, its non-degenerate simplices) is depicted by

Here, $x,y,z\in X_0$ are the $0$-th simplices defined by $x = d_1(f) = d_1(h)$, $y = d_0f$ and $z = d_0(h)$.

In particular, the horns themselves are simplicial subsets $\Lambda^n_k\hookrightarrow \Delta^n$, characterized by the sequence $(\delta^0,\dots,\delta^{k-1},\bullet,\delta^{k+1},\dots,\delta^n)$, where $\delta^i\in\Delta^n_{n-1}$ are the cofaces.