On graded and shifted notions, and thick morphisms

We consider the notions of $L_{\mathrm{\infty }}$-, $P_{\mathrm{\infty }}$-, and $S_{\mathrm{\infty }}$-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 $S_{\mathrm{\infty }}$-thick morphism” (which we introduce here) induces an $L_{\mathrm{\infty }}$-morphism of shifted $S_{\mathrm{\infty }}$-structures. The same holds for “shifted $P_{\mathrm{\infty }}$-thick morphisms” and shifted $P_{\mathrm{\infty }}$-structures, respectively.