  We construct so called Hall monoidal categories (and Hall modules thereover)
and exhibit them as a categorification of classical Hall and Hecke algebras
(and certain modules thereover). The input of the (functorial!) construction
are simplicial groupoids satisfying the $2$-Segal conditions (as introduced by
Dyckerhoff and Kapranov), the main examples come from Waldhausen's
S-construction. To treat the case of modules, we introduce a relative version
of the $2$-Segal conditions.
  Furthermore, we generalize a classical result about the representation theory
of symmetric groups to the case of wreath product groups: We construct a
monoidal equivalence between the category of complex $G\wr S_n$-representations
(for a fixed finite group $G$ and varying $n\in\mathbb N$) and the category of
"$G$-equivariant" polynomial functors; we use this equivalence to prove a
version of Schur-Weyl duality for wreath products.
