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  "body": "\n## Summary\n\nA 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.\nIt is the universal-algebra form of a [[section and retraction|DXJH3DGXBC]] in a category.\nRetracts are used to describe projective algebras: in a variety, the projective algebras are the retracts of free algebras.\n\n## Explanations\n\n### L1 human\n\nThink of a subalgebra as a piece inside a bigger algebra.\nIt is a retract when the whole algebra can be folded onto the piece, respecting the operations, without moving the points of the piece.\nIn 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.\nThe subgroup $2\\Z$ of $\\Z$ is not a retract, since no homomorphism $\\Z \\to 2\\Z$ fixes $2$.\n",
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