Cohomology and homotopy
In higher category theory
The global sections of a bundle are simply its sections. When bundles are replaced by their sheaves of local sections, then forming global sections corresponds to the direct image operation on sheaves with respect to the morphism to the terminal site. This definition generalizes to objects in a general topos and (∞,1)-topos.
We start describing the more explicit notions of global sections of bundles and then work our way towards the more abstract notions in terms of topos theory.
A global section of a bundle is simply a section of , that is a map such that .
The adjective ‘global’ here is used to distinguish from a local section: a generalised section over some subspace which is a section of the map to
from the pullback
Compare the notion of global point, which is really the special case when is a terminal object (where the generalised section corresponds to a generalised element). On the other hand, a global section of in is simply a global point in the slice category .
One often writes
for the set of global sections over (or or similar).
Of sheaves on topological spaces
Every sheaf on (the site that is given by the category of open subsets of) a topological space is the sheaf of local sections of its etale space bundle in that
for every . For this reasons one often speaks of the value of a sheaf on some object as a set of sections, even if the corresponding bundle is never mentioned and doesn’t really matter.
The set of global sections on is
where denotes the terminal object of the category of sheaves . Often this is written just using different notation
One notices that defined this way is the direct image functor on Grothendieck toposes that is induced from the canonical morphism of topological spaces (now “” really denotes the point topological space!) and hence from the corresponding morphism of sites.
Again, this expression for global sections induces a relative version, e.g. for sheaves on -schemes, the direct image functor goes into the base scheme ).
Of objects in a general Grothendieck topos
The definition of global sections of sheaves on topological spaces in terms of the direct image of the canonical morphism to the terminal site generalizes to sheaf toposes over arbitrary sites.
For every Grothendieck topos , there is a geometric morphism
called the global sections functor. It is given by the hom-set out of the terminal object
and hence assigns to each object its set of global elements .
The left adjoint of the global section functor is the canonical Set-tensoring functor
applied to the terminal object
which sends a set to the coproduct of copies of the terminal object
This is called the constant object of on the set . Notably when is a sheaf topos this is the constant sheaf on .
If the topos is a locally connected topos then the left adjoint functor is also a right adjoint, its left adjoint being the functor that sends an object to its set of connected components.
Of objects in an -topos
The previous abstract definition generalizes straightforwardly to every context of higher category theory where the required notions of adjoint functor etc. are provided.
Notably in (∞,1)-category theory the global section functor on an ∞-stack (∞,1)-topos is the hom-functor
of morphisms out of the terminal object.
This is indeed again the terminal geometric morphism
If the (∞,1)-topos is a locally contractible (∞,1)-topos then this is an essential geometric morphism.
The composite (∞,1)-functor is the shape of .