On graded and shifted notions, and thick morphisms

Abstract

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.