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https://en.wikipedia.org/wiki/Sheaf_(mathematics)
Sheaf (mathematics)
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In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one.
For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical.
They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category.
On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.

History
The first origins of sheaf theory are hard to pin down ? they may be co-extensive with the idea of analytic continuation[clarification needed]. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.

1936 Eduard ?ech introduces the nerve construction, for associating a simplicial complex to an open covering.
1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined cochains.
1943 Norman Steenrod publishes on homology with local coefficients.
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