---
tag: 7JS0JFR89E
kind: concept
name: Diffeology
msc: [58A40, 58A03]
sources:
  - url: https://en.wikipedia.org/wiki/Diffeology
same_as:
  wikidata: Q5275263
  wikipedia: Diffeology
---

## Summary

A diffeology on a set declares which maps from open subsets of Euclidean spaces into the set count as smooth; these maps are called plots.
A set with a diffeology is a diffeological space.
Every [[smooth manifold|F45VQ47GPV]] is a diffeological space, but so are quotients, subsets and spaces of smooth maps, which are often not manifolds.
Diffeological spaces form a [[cartesian closed category|BFBNQ36QS8]] with all limits and colimits, so constructions that leave the world of manifolds stay inside this one.

## Explanations

### L1 human

On a manifold, one knows which curves and surfaces inside it are smooth.
A diffeology takes this knowledge as the definition: a space is "smooth" once one says which parametrized families of points in it are smooth.
For instance, the quotient of the torus by a line of irrational slope has a useless topology, since its only open sets are the empty set and everything.
Yet its diffeology still remembers a lot of geometry.
This makes diffeology a good language for singular and infinite-dimensional spaces.