  These are some speculations on how Grothendieck's point of view and the idea
of complexity dynamics can come together in the problem of explaining the black
hole information paradox. They are neither complete, nor final, but can seem
like a new direction of research. If read as such they could prove useful to
some researchers. The basic idea is that entanglement alone cannot fully
account for the information extraction in black hole contexts. Complexity has
been proposed as an alternative but remains a vague concept. I employ
Grothendieck's point of view to expand the idea of entanglement entropy to a
categorical context in which the objects (states) and their maps are considered
together and the map space has additional topological and geometric structure
that intermingles with the object set of the category via Sieves, Sheafs, and
Toposes.
