I have two questions:
- In the nlab entry about the sweedler notation it reads "One can formalize in fact which manipulations are allowed with such a reduced notation." Has anyone done that/an idea on what kind of limitations there are? The article has no sources.
- Given a coalgebra $(A, \Delta_A, \epsilon; k)$, $k$ unital ring and a (left) $A$-comodule $\rho^M : M \to A \otimes_k M$ one writes $$ \rho^M(m) = \sum\limits_{(m)} m_{(-1)} \otimes m_{(0)}. $$ Now Hazewinkel et. al. [p. 278] write $$ \sum\limits_{(m)} m_{(-2)} \otimes m_{(-1)} \otimes m_{(0)} := (\Delta_A \otimes M) \rho^M(m) = (M \otimes \rho^M) \rho^M(m) $$ for the comodule axiom (which is the rightmost equation). This can be written as $$ \sum\limits_{(m),(m_{(-1)})} m_{(-1)_{(-2)}} \otimes m_{(-1)_{(-1)}} \otimes m_{(0)} = \sum\limits_{(m),(m_{(0)})} m_{(-1)} \otimes m_{(0)_{(-1)}} \otimes m_{(0)_{(0)}} $$ or $$ \sum\limits_{(m),(m_{(-1)})} m_{(-1)_{(1)}} \otimes m_{(-1)_{(2)}} \otimes m_{(0)} = \sum\limits_{(m),(m_{(0)})} m_{(-1)} \otimes m_{(0)_{(-1)}} \otimes m_{(0)_{(0)}}, $$ where the "stacked" indices are different. I haven't found anyone doing this computation. What should I use/is the right one? On one hand, the second expression uses the established comultiplication notation for coalgebras, i.e. $$ \Delta_A(a) = \sum\limits_{(a)} a_{(1)} \otimes a_{(2)}.$$ On the other hand, mixing the ordering of positive/negative integers (sparing the 0 for comodule "elements") and arriving at the (-2),(-1),(0) sequence suggests using the first one. Anyway, the problem obviously stems from defining the the comultiplication once, having both left and right comodules and trying to distinguesh them.