---
tag: 1FAF2VNA4G
kind: concept
name: CAT(k) space
msc: [53C23, 51F99]
specializes: [0VKD1TP7F3]
sources:
  - url: https://en.wikipedia.org/wiki/CAT(k)_space
same_as:
  wikidata: Q5008891
  wikipedia: CAT(k) space
---

## Summary

A CAT($k$) space is a [[metric space|0VKD1TP7F3]] whose geodesic triangles are at least as thin as triangles in a model surface.
The model surface has constant curvature $k$.
It is the metric version of the condition "sectional curvature at most $k$".
The notion applies to spaces with no smooth structure, such as trees, polyhedral complexes and buildings.
CAT(0) spaces are central in geometric group theory.

## Explanations

### L1 human

Draw a triangle in the space whose sides are shortest paths.
Draw a triangle with the same side lengths on a flat sheet, on a sphere or on a saddle, according to $k$.
The space is CAT($k$) when the first triangle is never fatter than the model one: points on its sides are no farther apart than the matching points of the model.
A tree is CAT(0): every triangle in a tree is a tripod, as thin as possible.
A round sphere is not CAT(0), because its triangles bulge outward.