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.
Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry.
First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space.
In such contexts, several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves.
Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces.
The ability to restrict data to smaller open subsets gives rise to the concept of presheaves.
A sheaf is a presheaf whose sections are, in a technical sense, uniquely determined by their restrictions.
It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space.
This observation is used to construct another example which is crucial in algebraic geometry, namely quasi-coherent sheaves.
is not a sheaf, since there is, in general, no way to preserve this property by passing to a smaller open subset.
Instead, this forms a cosheaf, a dual concept where the restriction maps go in the opposite direction than with sheaves.
In addition to the constant presheaf mentioned above, which is usually not a sheaf, there are further examples of presheaves that are not sheaves: One of the historical motivations for sheaves have come from studying complex manifolds,[4] complex analytic geometry,[5] and scheme theory from algebraic geometry.
One of the main historical motivations for introducing sheaves was constructing a device which keeps track of holomorphic functions on complex manifolds.
Sheaves are a direct tool for dealing with this complexity since they make it possible to keep track of the holomorphic structure on the underlying topological space of
For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks.
This categorical situation is the reason why the sheafification functor appears in constructing cokernels of sheaf morphisms or tensor products of sheaves, but not for kernels, say.
The stalk is an essential special case of the pullback in view of a natural identification, where
Due to its nice behavior on stalks, the extension by zero functor is useful for reducing sheaf-theoretic questions on
The property of being a locally ringed space translates into the fact that such a function, which is nonzero at a point
Like for modules, coherence is in general a strictly stronger condition than finite presentation.
In algebraic geometry, the natural analog of a covering map is called an étale morphism.
For example, a partition of unity argument shows that the sheaf of smooth functions on a manifold is soft.
Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space.
For example, computing the coherent sheaf cohomology of projective plane curves is easily found.
One big theorem in this space is the Hodge decomposition found using a spectral sequence associated to sheaf cohomology groups, proved by Deligne.
This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space and grassmann manifolds.
Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points.
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.
At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to algebraic topology.