horn of a simplex

Question

I'm reading one book of simplicial homotopy. It's just amazing. But I am stuck at the very beginning of the book. He let a simplicial set be a contravariant functor between the category $ \Delta $ of finite ordered sets $ \left\{ 1, ..., n\right\} $. Then he defines $ \Delta ^n $ to be the contravariant representable functor (with representation object being $ \left\{ 1,\ldots , n\right\} $ of the category $ \Delta $.

The horn of $ \Delta ^n $ is a subsimplex. But I didn't understand how to see it - he defines and visualizes it using the realization of $ \Delta ^n $. I would like to understand what is the boundary and the horn of $ \Delta ^n $ knowing that $ \Delta ^n (m) = Hom (m, n) $. Is there a nice way to see fast what is the horn in this language?

Answer 1

The combinatorial description of horns is equally straightforward: $\Lambda^n_k$ is the smallest simplicial subset such that $\Lambda^n_k ([n - 1])$ contains $\delta^i : [n - 1] \to [n]$ for all $i \ne k$. Equivalently, it is the union of the images of the simplicial maps $\delta^i : \Delta^{n-1} \to \Delta^n$ for all $i \ne k$.

Answer 2

A quick correction: in the category of finite ordered sets, "$n$" represents the ordered set $\{0,1,\ldots,n\}$ (which can be thought of as the vertices of $\Delta^n$).

In simplicial degree $m$, $\Lambda^n_k(m)$ is the subset of $\Delta^n(m) = Hom(m,n)$ consisting of those functions $f: m \to n$ whose image does not contain all of $$ \left\{0,1,\ldots, \widehat k,\ldots, n\right\}= \{0,1,\ldots,n\} \setminus \{k\}. $$