This paper is a slightly revised version of the author's thesis (November 1993) and is identical to the Heft 12 of the Schriftenreihe des Mathematischen Instituts Muenster.
Chapter I (chapter1.dvi) gives partial results on what we call the "generic relatively unipotent sheaf", which is defined for any sufficiently nice morphism with a section of schemes over C (in the Hodge theoretic context) or a number field (in the context of l-adic sheaves, or systems of smooth sheaves).
We then give the general formalism of the construction of polylogarithmic extensions (indeed, in the higher dimensional case, these extensions won't be one-extensions, hence can't be thought of as framed sheaves).
In complexity theory, the class NC ("Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.
The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor.
Equivalently, NC can be defined as those decision problems decidable by uniform Boolean circuits[?] with polylogarithmic depth and a polynomial number of gates.
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