# Contents

## Idea

The horn ${\Lambda }_{k}\left[n\right]={\Lambda }_{k}^{n}↪{\Delta }^{n}$ 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

$\Delta \left[n\right]=\Delta \left(-,\left[n\right]\right)\in \mathrm{Simp}\mathrm{Set}$\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 $\Delta \left[n\right]$, a subsimplicial set, ${\Lambda }^{i}\left[n\right]$, called the $\left(n,i\right)$-horn or $\left(n,i\right)$-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 ${\Lambda }^{k}\left[n\right]:{\Delta }^{\mathrm{op}}\to \mathrm{Set}$ must therefore be: it takes $\left[m\right]$ to the collection of ordinal maps $f:\left[m\right]\to \left[n\right]$ which do not have the element $k$ in the image.

The horn ${\Lambda }^{k}\left[n\right]$ is an outer horn if $k=0$ or $k=n$. Otherwise it is an inner horn.

## Examples

The inner horn of the 2-simplex

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

with boundary

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

looks like

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

The two outer horns look like

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

and

${\Lambda }_{2}^{2}=\left\{\begin{array}{ccc}& & 1\\ & & & ↘\\ 0& & \to & & 2\end{array}\right\}$\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 ${\Lambda }^{k}\left[n\right]↪{\Delta }^{n}$.

• 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.

Revised on June 25, 2012 01:20:00 by Andrew Stacey (80.203.115.55)