nLab
horn

Idea

The horn ${\Lambda }_k[n]={\Lambda }_k^n\hookrightarrow {\Delta }^n$ 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.

Definition

Let

$$\Delta [n]=\Delta (-,[n])\in \mathrm{Simp}\mathrm{Set}$$\Delta[n] = \mathbf{\Delta}( -, [n]) \in Simp Set

be the standard simplicial $n$-simplex in SimpSet.

Then, for each $i$, $0\le i\le n$, 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 $i^{\mathrm{th}}$ one.

Since $\mathrm{SimpSet}$ 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 ${\Lambda }^k[n]:{\Delta }^{\mathrm{op}}\to \mathrm{Set}$ 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.

Examples

The inner horn of the 2-simplex

$${\Delta }^2=\left\{\begin{array}{ccc}& & 1\\ & \nearrow & \Downarrow & \searrow \\ 0& & \to & & 2\end{array}\right\}$$\Delta^2 = \left\{ \array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \right\}

with boundary

$$\partial {\Delta }^2=\left\{\begin{array}{ccc}& & 1\\ & \nearrow & & \searrow \\ 0& & \to & & 2\end{array}\right\}$$\partial \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\}

looks like

$${\Lambda }_1^2=\left\{\begin{array}{ccc}& & 1\\ & \nearrow & & \searrow \\ 0& & & & 2\end{array}\right\}$$\Lambda^2_1 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\}

The two outer horns look like

$${\Lambda }_0^2=\left\{\begin{array}{ccc}& & 1\\ & \nearrow & & \\ 0& & \to & & 2\end{array}\right\}$$\Lambda^2_0 = \left\{ \array{ && 1 \\ & \nearrow && \\ 0 &&\to&& 2 } \right\}

and

$${\Lambda }_2^2=\left\{\begin{array}{ccc}& & 1\\ & & & \searrow \\ 0& & \to & & 2\end{array}\right\}$$\Lambda^2_2 = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}

respectively.

Relation to other concepts

• 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.