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