  We consider a generalised model of a random simplicial complex, which arises
from a random hypergraph. Our model is generated by taking the downward-closure
of a non-uniform binomial random hypergraph, in which for each $k$, each set of
$k+1$ vertices forms an edge with some probability $p_k$ independently. As a
special case, this contains an extensively studied model of a (uniform) random
simplicial complex, introduced by Meshulam and Wallach [Random Structures &
Algorithms 34 (2009), no. 3, pp. 408-417].
  We consider a higher-dimensional notion of connectedness on this new model
according to the vanishing of cohomology groups over an arbitrary abelian group
$R$. We prove that this notion of connectedness displays a phase transition and
determine the threshold. We also prove a hitting time result for a natural
process interpretation, in which simplices and their downward-closure are added
one by one. In addition, we determine the asymptotic behaviour of cohomology
groups inside the critical window around the time of the phase transition.
