symmetric monoidal (∞,1)-category of spectra
Sweedler notation is a special notation for discussion of operations in coalgebras
If $C$ is a coassociative coalgebra and then for $c\in C$, the comultiplication $\Delta$ maps $c$ to an element in $C\otimes C$ which is therefore a sum of the form $\sum_{i=1}^n a_i\otimes b_i$. Sweedler suggests that we do not make up new symbols like $a$ and $b$ but rather use composed symbols $c_{(1)}$ and $c_{(2)}$. Therefore
Sweedler notation means that for certain manipulations involving just generic linear operations we actually do not need to think of the summation symbol $i$, so we can just write
with or even without summation sign. Surely in either case we need to remember that we do not have a factorization but we do have a sum of possibly more than one entry. One can formalize in fact which manipulations are allowed with such a reduced notation.
It gets more useful, when we take into account coassociativity to justify extending the notation to write
Furthermore, we can extend it to coactions, e.g. $\rho:V\to V\otimes C$, by $\rho(v) = \sum v_{(0)}\otimes v_{(1)}$. Then we can use the coaction axiom $(id_V\otimes \Delta)\circ\rho = (\rho\otimes id_C)\circ \rho$ to write
where we used the sumless Sweedler notation.
One big use is that the scalars like $\epsilon(a_{(3)})$ can be moved freely along the expression, which is difficult to write without calculating with Sweedler components: one would need lots of brackets and flip operators, and this could be messy and abstract.
The notation is named after Moss Sweedler. Sometimes (though rarely) it is also called Heyneman-Sweedler notation.