Editing Coalgebra (section)

== Sweedler notation ==
When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element ''c'' of the coalgebra (''C'', Δ, ε), there exist elements ''c''{{su|b=(1)|p=(''i''&hairsp;)}} and ''c''{{su|b=(2)|p=(''i''&hairsp;)}} in ''C'' such that 
:<math>\Delta(c)=\sum_i c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}</math>
Note that neither the number of terms in this sum, nor the exact values of each <math>c_{(1)}^{(i)}</math> or <math>c_{(2)}^{(i)}</math>, are uniquely determined by <math>c</math>; there is only a promise that there are finitely many terms, and that the full sum of all these terms <math>c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}</math> have the right value <math>\Delta(c)</math>.

In ''Sweedler's notation'',<ref name=Und35>Underwood (2011) p.35</ref> (so named after [[Moss Sweedler]]), this is abbreviated to
:<math>\Delta(c)=\sum_{(c)} c_{(1)}\otimes c_{(2)}.</math>

The fact that ε is a counit can then be expressed with the following formula
:<math>c=\sum_{(c)} \varepsilon(c_{(1)})c_{(2)} = \sum_{(c)} c_{(1)}\varepsilon(c_{(2)}).\;</math>
Here it is understood that the sums have the same number of terms, and the same lists of values for <math>c_{(1)}</math> and <math>c_{(2)}</math>, as in the previous sum for <math>\Delta(c)</math>.

The coassociativity of Δ can be expressed as 
:<math>\sum_{(c)}c_{(1)}\otimes\left(\sum_{(c_{(2)})}(c_{(2)})_{(1)}\otimes (c_{(2)})_{(2)}\right) = \sum_{(c)}\left( \sum_{(c_{(1)})}(c_{(1)})_{(1)}\otimes (c_{(1)})_{(2)}\right) \otimes c_{(2)}.</math>
In Sweedler's notation, both of these expressions are written as
:<math>\sum_{(c)} c_{(1)}\otimes c_{(2)}\otimes c_{(3)}.</math>

Some authors omit the summation symbols as well; in this sumless Sweedler notation, one writes
:<math>\Delta(c)=c_{(1)}\otimes c_{(2)}</math>
and
:<math>c=\varepsilon(c_{(1)})c_{(2)} = c_{(1)}\varepsilon(c_{(2)}).\;</math>
Whenever a variable with lowered and parenthesized index is encountered in an expression of this kind, a summation symbol for that variable is implied.