nLab
flag

In geometry, a flag is a chain of incidence relations, as for example between distinct linear subspaces

$$V_0\subseteq V_1\subseteq \dots \subseteq V_n$$V_0 \subseteq V_1 \subseteq \ldots \subseteq V_n

of a fixed vector space $V$, or between isotropic subspaces, etc.

Flags of posets

Generally speaking, if $P$ is a poset, a flag is a chain

$$x_0<x_1<\dots <x_n$$x_0 \lt x_1 \lt \ldots \lt x_n

and the set of elements $\{x_0,x_1,\dots ,x_n\}$ can be thought of as an $n$-simplex of a simplicial complex whose vertices are the poset elements. Hence we have a functor

$$\mathrm{Flag}:\mathrm{Pos}\to \mathrm{SimpComplex}$$Flag: Pos \to SimpComplex

In the other direction, there is an underlying functor

$$U:\mathrm{SimpComplex}\to \mathrm{Pos}$$U: SimpComplex \to Pos

which sends a simplicial complex $(V,\Sigma )$ to $\Sigma$ (regarded as a poset ordered by inclusion). The composite

$$\mathrm{Flag}\circ U:\mathrm{SimpComplex}\to \mathrm{SimpComplex}$$Flag \circ U: SimpComplex \to SimpComplex

is called the subdivision functor, or, more exactly, the barycentric subdivision functor.