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This is about the differential idea of a sheaf as a vector bundle $E$ (not necessarily locally trivial) on a real manifold $M$ with a zero curvature connection $\nabla$. The zeroth cohomology is then the sections of $E$ with zero derivative, and this extends to higher degrees. The things I am going to ask about have answers in terms of the pre sheaf / open set definition of sheaf, but I want to be more restrictive for the following reason: I am looking at non commutative geometry where there is currently no satisfactory idea of restriction to an open set, but there are established ideas of differential calculus. I am hoping that there are existing classical answers, so that some form of provisional non commutative definitions can be formed. I am aware that using this differential approach has disadvantages, but from a noncommutative setting there are limited options given the current state of the theory. Possibly the differential way of looking at sheaves is most familiar from algebraic geometry.

1) The skyscraper construction: Given a vector bundle $F$ over $\mathbb{R}$ with a zero curvature connection, form a similar construction for $\mathbb{R^2}$ for a bundle with support on the $y$-axis, and which is just $F$ when restricted to the $y$ axis. (Is this still a skyscraper sheaf when its support is a line? I decided that a point was too much of a special case to see what was happening. Probably not, but never mind.)

2) Is there a definition of a push forward sheaf for the differential definition above? Given a smooth map $f:M\to N$, what is the push forward of the construction $(M,E,\nabla)$ above?

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