  In a recent work of Galatius and Venkatesh, the authors showed the importance
of studying simplicial generalizations of Galois deformation functors. They
established a precise link between the simplicial universal deformation ring
$R$ prorepresenting such a deformation problem (with local conditions) and a
derived Hecke algebra. Here we focus on the algebraic part of their study which
we complete in two directions. First, we introduce the notion of simplicial
pseudo-characters and prove relations between the (derived) deformation
functors of simplicial pseudo-characters and that of simplicial Galois
representations. Secondly, we define the relative cotangent complex of a
simplicial deformation functor and, in the ordinary case, we relate it to the
relative complex of ordinary Galois cochains. Finally, we recall how the latter
can be used to relate the fundamental group of $R$ to the ordinary dual adjoint
Selmer group, by a homomorphism already introduced in Galatius-Venkatesh and
studied in greater generality in Tilouine-Urban.
