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  "body": "\n## Summary\n\nThe inverse limit of an inverse system of algebras is the algebra of all compatible families of elements.\nSuch a family has one element in each algebra of the system.\nIt is a subalgebra of the [[direct product|WJ73XYBNE4]] of the algebras.\nIt is the limit of the system in the category of algebras of the given signature.\nInverse limits build large algebras, such as the $p$-adic integers and profinite groups, out of a coherent family of smaller ones.\n\n## Explanations\n\n### L1 human\n\nPicture a tower of algebras, each one mapping to the one below it by a [[homomorphism|KBXSZC6BV1]].\nA point of the inverse limit is a choice of one element on every floor, such that each choice maps to the choice on the floor below.\nFor example, take the rings $\\Z/2\\Z$, $\\Z/4\\Z$, $\\Z/8\\Z$, and so on, with reduction maps between them.\nA compatible family such as $(1, 3, 7, 15, \\ldots)$ is an element of the inverse limit, which is the ring of $2$-adic integers.\n",
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