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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]].\nA *coalgebra* over $K$ is a [[vector space|000AYPA9AJ]] $C$ over $K$ with [[linear maps|0NRYHJPQG2]] $\\Delta \\colon C \\to C \\otimes C$, the comultiplication, and $\\varepsilon \\colon C \\to K$, the counit, such that:\n\n1. $(\\Delta \\otimes \\id_C) \\circ \\Delta = (\\id_C \\otimes \\Delta) \\circ \\Delta$ as maps $C \\to C \\otimes C \\otimes C$ (coassociativity).\n2. $(\\varepsilon \\otimes \\id_C) \\circ \\Delta = \\id_C = (\\id_C \\otimes \\varepsilon) \\circ \\Delta$, where $K \\otimes C$ and $C \\otimes K$ are identified with $C$ (counit law).\n\nHere $\\otimes$ is the [[tensor product|0BF2DE2SYK]] over $K$.\n\n## Remarks\n\nExample: for a set $S$, let $C = K[S]$ be the vector space with basis $S$, and put $\\Delta(s) = s \\otimes s$ and $\\varepsilon(s) = 1$ for $s \\in S$.\nBoth sides of clause 1 send $s$ to $s \\otimes s \\otimes s$, and both maps in clause 2 send $s$ to $1 \\cdot s = s$.\nNon-example: the same $\\Delta$ with $\\varepsilon(s) = 0$ for all $s$ fails clause 2, since $(\\varepsilon \\otimes \\id)\\Delta(s) = 0 \\ne s$.\nAnother example is $K[x]$ with $\\Delta(x^n) = \\sum_{k=0}^{n} \\binom{n}{k} x^k \\otimes x^{n-k}$ and $\\varepsilon(x^n) = 1$ if $n = 0$ and $0$ otherwise.\nThe definition works in the same way over a commutative ring in place of $K$.\nOver a field, every element of a coalgebra lies in a finite-dimensional subcoalgebra.\n",
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