# nLab flag

## Definition

### General

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

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

of a fixed vector space $V$, or between isotropic subspaces, etc. A flag is complete if $\dim V_i = i$ for each $i \in \{0, \ldots, n\}$.

### Flags of posets

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

$x_0 \lt x_1 \lt \ldots \lt x_n$

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

$Flag: Pos \to SimpComplex$

In the other direction, there is an underlying functor

$U: SimpComplex \to Pos$

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

$Flag \circ U: SimpComplex \to SimpComplex$

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

Last revised on October 22, 2019 at 00:24:13. See the history of this page for a list of all contributions to it.