nLab flag



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

V 0V 1V nV_0 \subseteq V_1 \subseteq \ldots \subseteq V_n

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

Flags of posets

Generally speaking, if PP is a poset, a flag is a chain

x 0<x 1<<x nx_0 \lt x_1 \lt \ldots \lt x_n

and the set of elements {x 0,x 1,,x n}\{x_0, x_1, \ldots, x_n\} can be thought of as an nn-simplex of a simplicial complex whose vertices are the poset elements. Hence we have a functor

Flag:PosSimpComplexFlag: Pos \to SimpComplex

In the other direction, there is an underlying functor

U:SimpComplexPosU: SimpComplex \to Pos

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

FlagU:SimpComplexSimpComplexFlag \circ U: SimpComplex \to SimpComplex

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

Last revised on October 22, 2019 at 04:24:13. See the history of this page for a list of all contributions to it.