# nLab flag

## Defintion

### General

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 $\left\{{x}_{0},{x}_{1},\dots ,{x}_{n}\right\}$ 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 $\left(V,\Sigma \right)$ 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.

Revised on September 4, 2013 14:41:22 by Urs Schreiber (212.238.84.235)