  This paper is motivated by questions such as P vs. NP and other questions in
Boolean complexity theory. We describe an approach to attacking such questions
with cohomology, and we show that using Grothendieck topologies and other ideas
from the Grothendieck school gives new hope for such an attack.
  We focus on circuit depth complexity, and consider only finite topological
spaces or Grothendieck topologies based on finite categories; as such, we do
not use algebraic geometry or manifolds.
  Given two sheaves on a Grothendieck topology, their "cohomological
complexity" is the sum of the dimensions of their Ext groups. We seek to model
the depth complexity of Boolean functions by the cohomological complexity of
sheaves on a Grothendieck topology. We propose that the logical AND of two
Boolean functions will have its corresponding cohomological complexity bounded
in terms of those of the two functions using ``virtual zero extensions.'' We
propose that the logical negation of a function will have its corresponding
cohomological complexity equal to that of the original function using duality
theory. We explain these approaches and show that they are stable under
pullbacks and base change. It is the subject of ongoing work to achieve AND and
negation bounds simultaneously in a way that yields an interesting depth lower
bound.
