Part I. Vladimir Salnikov, part II. Alexei Kotov
In this talk we introduce some natural constructions from the "graded world", paying particular attention to the differences between N- and Z- graded manifolds.
We start by the construction of the sheaf of functions on graded manifolds, and describe its structure. The intrinsic properties of this functional space are conveniently given using the language of filtrations, allowing to formulate the analog of Batchelor’s theorem.
Afterwards we introduce graded Hopf algebras and Harish-Chandra pairs, which in turn provide the result of equivalence of categories between graded Lie groups and algebras. These constructions are then used to solve the integration problem of differential graded Lie algebras to differential graded Lie groups.
This talk is based on:
 B. Jubin, A. Kotov, N. Poncin, V. Salnikov, Differential graded Lie groups and their differential graded Lie algebras, Transformation Groups, 27, 2022
 A. Kotov, V. Salnikov, The category of Z-graded manifolds: what happens if you do not stay positive, Preprint: arXiv:2108.13496
 A. Kotov, V. Salnikov, Various instances of Harish-Chandra pairs, Preprint: arXiv:2207.07083