The horn $\Lambda_k[n] = \Lambda^n_k \hookrightarrow \Delta^n$ is the simplicial set obtained from the boundary of the n-simplex $\partial \Delta^n$ of the standard simplicial $n$-simplex $\Delta^n$ by discarding the $k$th face.
Let
be the standard simplicial $n$-simplex in SimpSet.
Then, for each $i$, $0 \leq i \leq n$, we can form, within $\Delta[n]$, a subsimplicial set, $\Lambda^i[n]$, called the $(n,i)$-horn or $(n,i)$-box, by taking the union of all faces but the $i^{th}$ one.
Since $SimpSet$ is a presheaf topos, unions of subobjects make sense, they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor $\Lambda^k[n]: \Delta^{op} \to Set$ must therefore be: it takes $[m]$ to the collection of ordinal maps $f: [m] \to [n]$ which do not have the element $k$ in the image.
The horn $\Lambda^k[n]$ is an outer horn if $k = 0$ or $k = n$. Otherwise it is an inner horn.
The inner horn of the 2-simplex
with boundary
looks like
The two outer horns look like
and
respectively.
A Kan fibration is a morphism of simplicial sets which has the right lifting property with respect to all horn inclusions $\Lambda^k[n] \hookrightarrow \Delta^n$.
A Kan complex is a simplicial set in which “all horns have fillers“: a simplicial set for which the morphism to the point is a Kan fibration.
A quasi-category is a simplicial set in which all inner horns have fillers.
The boundary of a simplex is the union of its faces.
The spine of a simplex is the union of all its generating 1-cells.