A DGA has a multiplication like \begin{align*} xy = (-1)^{{\left\lvert {x} \right\rvert} {\left\lvert {y} \right\rvert}} yx .\end{align*} What if you replaced this with some kind of \(p{\hbox{-}}\)graded condition instead? Note that \(\left\{{\pm 1}\right\} = \mu_2\), so maybe define a character \(\chi_2: A{ {}^{ \scriptstyle\otimes_{k}^{2} } } \to \mu_2\) to rewrite this as \begin{align*} xy = \chi_2(x, y) yx .\end{align*} More generally, could you define something like \(\chi_p: A{ {}^{ \scriptstyle\otimes_{k}^{2} } } \to \mu_p\)?

Super super vague! Inspired by the idea that certain maps \(f\) between DGAs don’t commute with differentials, so aren’t DGA morphisms, but do satisfy \({{\partial}}f(\omega) = p f({{\partial}}\omega)\) instead. See p-derivation?

#web/quick_notes #personal/idle-thoughts