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.

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