---
tag: J32951135H
kind: concept
name: Order complex
msc: [06A07, "05E45", 55U10]
sources:
  - url: https://en.wikipedia.org/wiki/Poset_topology
---

## Summary

The order complex of a [[partially ordered set|0QV2JS9RJ3]] $P$ is a [[simplicial complex|V8GZ0W1ZZ5]] with the elements of $P$ as vertices.
Its faces are the finite [[chains|0S8FE7ZMFM]] of $P$.
It turns a poset into a topological space, its [[geometric realization|TG6WHHR7CF]].
So one can speak of the homotopy type and the homology of a poset.
This is the basis of poset topology.
For example, take an open interval of a finite poset.
By a theorem of Philip Hall, the reduced Euler characteristic of its order complex equals the value of the [[Möbius function|2Y75S8NG5B]] on that interval.
The order complex of the face poset of a simplicial complex is its [[barycentric subdivision|J5DZEYCEX2]].

## Explanations

### L1 human

Draw a point for each element of the poset.
Join two points by an edge when they are comparable, fill in a triangle for each chain of three elements, a tetrahedron for each chain of four, and so on.
The resulting shape is the order complex.
For a chain $a < b < c$ the shape is a full triangle.
For the poset with $a, b < c, d$ and no other relations, the four edges $ac$, $ad$, $bc$, $bd$ form a square loop, which is a circle.