---
tag: 68PFSGP39B
kind: concept
name: Retract
msc: [08A30, 08A35]
sources:
  - url: https://en.wikipedia.org/wiki/Section_(category_theory)
---

## Summary

A retract of an [[algebra|QJGPT2W0FT]] is a [[subalgebra|7CHG6SD365]] onto which the whole algebra can be mapped by a [[homomorphism|KBXSZC6BV1]] that fixes the subalgebra pointwise.
It is the universal-algebra form of a [[section and retraction|DXJH3DGXBC]] in a category.
Retracts are used to describe projective algebras: in a variety, the projective algebras are the retracts of free algebras.

## Explanations

### L1 human

Think of a subalgebra as a piece inside a bigger algebra.
It is a retract when the whole algebra can be folded onto the piece, respecting the operations, without moving the points of the piece.
In the group $\Z \times \Z$, the subgroup $\Z \times \set{0}$ is a retract: the map $(m, n) \mapsto (m, 0)$ folds everything onto it.
The subgroup $2\Z$ of $\Z$ is not a retract, since no homomorphism $\Z \to 2\Z$ fixes $2$.