nLab
subdivision

Idea

Subdivision is a usually functorial process which takes as input some combinatorial notion of space (for example, a simplicial complex or simplicial set) and produces as output a more finely meshed space. It is related to the notion of classical subdivision.

Definition and Properties

For simplicial complexes

Subdivision is easiest to define for simplicial complexes. We have a pair of functors

$$\mathrm{SimpComplex}\stackrel{\stackrel{U}{\to }}{\underset{\mathrm{Flag}}{\leftarrow }}\mathrm{Pos}$$SimpComplex \stackrel{\overset{U}{\to}}{\underset{Flag}{\leftarrow}} Pos

where

• The functor $U$ sends a simplicial complex $(V,\Sigma )$ to $\Sigma$, regarded as a poset ordered by inclusion, and

• The functor $\mathrm{Flag}$ sends a poset $P$ to the simplicial complex whose vertex set is $P$ and whose simplices are the underlying sets $\{x_0,\dots ,x_n\}$ of flags $x_0<\dots <x_n$.

The composite $\mathrm{Flag}\circ U$ is called the subdivision $\mathrm{Sd}$; it is an endofunctor of $\mathrm{SimpComplex}$. Note that the vertices of $\mathrm{Sd}(X)$ are the simplices of $X$. We also have $\mid \mathrm{Sd}(X)\mid \cong \mid X\mid$, where $\mid -\mid$ is the usual geometric realization of simplicial complexes.

For simplicial sets

We can now define the subdivision of a simplicial set as follows. Recall that we can make any simplicial complex into a simplicial set. If we do this for the standard $n$-simplex, which as a simplicial complex is the set $([n],P([n])$ for $[n]=\{0,1,\dots ,n\}$, then we get the standard $n$-simplex simplicial set ${\Delta }^n$. We can then define the subdivision of ${\Delta }^n$, as a simplicial set, to be the simplicial set corresponding to its simplicial-complex subdivision. Finally, we can make this functorial on maps between standard simplices, and left Kan extend to a cocontinuous endofunctor of $\mathrm{SSet}$.

Applications

By construction (or the adjoint functor theorem), the subdivision functor $\mathrm{Sd}$ on simplicial sets has a right adjoint, called $\mathrm{Ex}$. An infinity iteration of $\mathrm{Ex}$ can be used to construct Kan fibrant replacements in the model structure on simplicial sets.

This functorial subdivision corresponds to the classical barycentric subdivision. Other classical subdivisions that are frequently encountered include the middle edge subdivision. This latter is closely related to the ordinal subdivision? of simplicial sets.