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  "body": "\n## Definition\n\nLet $K$ be a field and let $(C, \\Delta, \\varepsilon)$ be a [[coalgebra|A06RA8KB0T]] over $K$.\nFor $c \\in C$, the element $\\Delta(c) \\in C \\otimes C$ can be written as a finite sum $\\sum_{i=1}^{r} a_i \\otimes b_i$ with $a_i, b_i \\in C$.\n*Sweedler notation* writes this as\n$$\\Delta(c) = \\sum c_{(1)} \\otimes c_{(2)},$$\nand an expression $\\sum f(c_{(1)}, c_{(2)})$ means $\\sum_{i} f(a_i, b_i)$ for a map $f$ that is $K$-bilinear in its two arguments.\nBy coassociativity, the element $(\\Delta \\otimes \\id)\\Delta(c) = (\\id \\otimes \\Delta)\\Delta(c)$ is written $\\sum c_{(1)} \\otimes c_{(2)} \\otimes c_{(3)}$.\nFor a right [[comodule|7WEM2E6RTV]] $M$ with coaction $\\rho$, one writes $\\rho(m) = \\sum m_{(0)} \\otimes m_{(1)}$, with $m_{(0)} \\in M$ and $m_{(1)} \\in C$.\n\n## Remarks\n\nBecause $f$ is bilinear, $\\sum_i f(a_i, b_i)$ depends only on $\\Delta(c)$ and not on the chosen sum; it equals $\\tilde f(\\Delta(c))$ for the linear map $\\tilde f$ on $C \\otimes C$ induced by $f$.\nExample: the counit law reads $\\sum \\varepsilon(c_{(1)}) c_{(2)} = c = \\sum c_{(1)} \\varepsilon(c_{(2)})$.\nExample: for a group-like element $g$, with $\\Delta(g) = g \\otimes g$, one may take $g_{(1)} = g_{(2)} = g$.\nNon-example: $\\sum c_{(1)} \\otimes c_{(1)}$ is not a valid expression, because it is not bilinear in $(c_{(1)}, c_{(2)})$.\nFor $\\Delta(x) = x \\otimes 1 + 1 \\otimes x$ in $K[x]$ it would give $x \\otimes x + 1 \\otimes 1$.\nThe same element written as $(x + 1) \\otimes 1 + 1 \\otimes x - 1 \\otimes 1$ would give $(x+1) \\otimes (x+1) + 1 \\otimes 1 + 1 \\otimes 1$, a different result.\n",
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