We consider the notions of $L_{ \infty}$-, $P_{ \infty}$-, and $S_{ \infty}$-algebras (including "shifted" versions) in the $\mathbb{ Z}_2 \times \mathbb{ Z}$-graded setting. We also consider thick (microformal) morphisms and show how they work in such graded context. In particular, we show that a "shifted $S_{ \infty}$-thick morphism" (which we introduce here) induces an $L_{ \infty}$-morphism of shifted $S_{ \infty}$-structures. The same holds for "shifted $P_{ \infty}$-thick morphisms" and shifted $P_{ \infty}$-structures, respectively.
Categories: math.RA math-ph math.DG math.MP math.QA