  We prove relative injectivity, torsion-freeness, and vanishing theorems for
generalized normal crossing pairs on schemes, algebraic stacks, formal schemes,
semianalytic germs of complex analytic spaces, rigid analytic spaces, Berkovich
spaces, and adic spaces locally of weakly finite type over a field, all in
equal characteristic zero. We give a uniform proof for all these theorems in
all the categories of spaces mentioned above, which were previously only known
for varieties and complex analytic spaces due to work of Ambro and Fujino.
Ambro and Fujino's results are integral in the proofs of the fundamental
theorems of the minimal model program for (semi-)log canonical pairs and the
theory of quasi-log structures. Our results resolve a significant barrier to
extending these results on (semi-)log canonical pairs and quasi-log structures
beyond the setting of varieties and complex analytic spaces.
  In order to prove our most general injectivity theorems, we generalize to all
these categories of spaces a criterion due to Guill\'en and Navarro Aznar
characterizing when functors defined on smooth varieties extend to all
varieties. This extension result uses cubical hyperresolutions, which we
construct in all categories of spaces mentioned above. Our extension result is
very general and is of independent interest. We use this extension result to
prove our injectivity theorems for generalized normal crossing pairs. We also
apply our extension result to develop the theoretical foundations for the
Deligne-Du Bois complex in these categories of spaces and to construct a weight
filtration on the (pro-)\'etale cohomology of schemes and rigid analytic
spaces. These results establish some aspects of Deligne-Hodge theory in all
categories of spaces mentioned above.
