---
tag: E2870TEDG7
kind: concept
name: Weak equivalence
msc: [18N40, 55U35]
sources:
  - url: https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)
same_as:
  wikidata: Q20878207
  wikipedia: Weak equivalence (homotopy theory)
---

## Summary

In abstract homotopy theory, the weak equivalences of a category are the morphisms that one wants to treat as if they were isomorphisms.
They form one of the three classes of a model category, and the homotopy category is obtained by inverting them.
The motivating examples are the [[weak homotopy equivalences|4Q3WY1RX91]] of topological spaces and the [[quasi-isomorphisms|C9JEC2WQKV]] of chain complexes.
A weak equivalence need not have an inverse morphism, which is why the word weak is used.

## Explanations

### L1 human

Two spaces can have the same holes in every dimension without there being a nice map back and forth between them.
A weak equivalence is a map that sees no difference between its source and target from the chosen point of view, for example in all homotopy groups.
Homotopy theory then declares such a map to be as good as an isomorphism.
For chain complexes, a map that induces isomorphisms on all homology groups is a weak equivalence, even when it is far from invertible.
An example is the map from the complex $\Z \xrightarrow{2} \Z$, placed in degrees one and zero, to the complex with $\Z/2$ in degree zero and $0$ elsewhere.