nLab
boundary of a simplex

Contents

Idea

In as far as the simplicial nn-simplex Δ n\Delta^n (a simplicial set) is a combinatorial model for the nn-ball, its boundary Δ n\partial \Delta^n is a combinatorial model for the (n1)(n-1)-sphere.

Definition

The boundary Δ n\partial \Delta^n of the simplicial nn-simplex Δ n\Delta^n is the simplicial set generated from the simplicial set Δ n\Delta^n minus its unique non-degenerate cell in dimension nn.

This may equivalently be described to be degreewise the coequalizer

0i<jnΔ[n2] 0inΔ[n1]Δ[n] \coprod_{0\le i\lt j\le n}\Delta[n-2]\rightrightarrows\coprod_{0\le i\le n}\Delta[n-1]\to \partial \Delta[n]

defined by the (induced coproduct maps of the) simplicial identities d id j=d j1d id_i d_j=d_{j-1} d_i.

Regarding Δ n\Delta^n as the presheaf on the simplex category that is represented by [n]Obj(Δ)[n] \in Obj(\Delta), then this means that Δ n\partial \Delta^n is the simplicial set generated from Δ\Delta minus the identity morphism Id [n]Id_{[n]}.

There is a canonical monomorphism

i n:Δ nΔ n, i_n : \partial \Delta^n \hookrightarrow \Delta^n \,,

the boundary inclusion .

The geometric realization of this is the inclusion of the (n1)(n-1)-sphere as the boundary of the nn-disk.

Simplicial boundary inclusions are one part of the cofibrant generation of the classical model structure on simplicial sets.

Examples

For low nn the boundaries of nn-simplices look as follows (see also the illustrations at oriental)

  • Δ 0=\partial \Delta^0 = \emptyset;

  • Δ 1={01}={0,1}=Δ 0Δ 0\partial \Delta^1 = \partial\{0 \to 1\} = \{0, 1\} = \Delta^0 \sqcup \Delta^0;

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

Revised on March 12, 2012 21:14:27 by Urs Schreiber (89.204.153.191)