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  "body": "\n## Summary\n\nA coalgebra is a vector space $C$ over a field $K$ with a comultiplication $\\Delta \\colon C \\to C \\otimes C$ and a counit $\\varepsilon \\colon C \\to K$.\nIts axioms come from reversing the arrows in the definition of an [[associative algebra|3CTW1P4HBP]].\nThe dual space of a coalgebra is an algebra, and the dual of a finite-dimensional algebra is a coalgebra.\nCoalgebras are the other half of [[bialgebras|Q9HHVN606G]] and [[Hopf algebras|7XFYP0N84H]], and they model ways of splitting an object into two pieces.\n\n## Explanations\n\n### L1 human\n\nMultiplication takes two elements and returns one; comultiplication takes one element and returns a sum of pairs.\nIt lists the ways to split an element into two parts.\nFor the polynomial ring $K[x]$, one can split $x^n$ as $\\Delta(x^n) = \\sum_{k} \\binom{n}{k} x^k \\otimes x^{n-k}$, like choosing which of $n$ objects go to the left.\nThe counit picks out the part where nothing is split off.\nCoassociativity says that splitting into three parts gives the same result whichever way one splits first.\n",
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