nLab
horn

Contents

Idea

The horn Λ k[n]=Λ k nΔ n\Lambda_k[n] = \Lambda^n_k \hookrightarrow \Delta^n is the simplicial set obtained from the boundary of the n-simplex Δ n\partial \Delta^n of the standard simplicial nn-simplex Δ n\Delta^n by discarding the kkth face.

Definition

Let

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

be the standard simplicial nn-simplex in SimpSet.

Then, for each ii, 0in0 \leq i \leq n, we can form, within Δ[n]\Delta[n] , a subsimplicial set, Λ i[n]\Lambda^i[n], called the (n,i)(n,i)-horn or (n,i)(n,i)-box, by taking the union of all faces but the i thi^{th} one.

Since SimpSetSimpSet 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 Λ k[n]:Δ opSet\Lambda^k[n]: \Delta^{op} \to Set must therefore be: it takes [m][m] to the collection of ordinal maps f:[m][n]f: [m] \to [n] which do not have the element kk in the image.

The horn Λ k[n]\Lambda^k[n] is an outer horn if k=0k = 0 or k=nk = n. Otherwise it is an inner horn.

Examples

The inner horn of the 2-simplex

Δ 2={ 1 0 2}\Delta^2 = \left\{ \array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \right\}

with boundary

Δ 2={ 1 0 2}\partial \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\}

looks like

Λ 1 2={ 1 0 2} \Lambda^2_1 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\}

The two outer horns look like

Λ 0 2={ 1 0 2}\Lambda^2_0 = \left\{ \array{ && 1 \\ & \nearrow && \\ 0 &&\to&& 2 } \right\}

and

Λ 2 2={ 1 0 2}\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 Λ k[n]Δ n\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.

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