(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
For a geometry (for structured (∞,1)-toposes) a -structured (∞,1)-topos is locally representable if it is locally equivalent to for (the pro-objects in an (∞,1)-category), or itself if it is locally finite presented .
Let be a geometry (for structured (∞,1)-toposes). Write for the underlying discrete geometry. The identity functor
is then a morphism of geometries.
Recall the notation for the (∞,1)-category of -structured (∞,1)-toposes and geometric morphisms between them.
Theorem ( StSp 2.1.1 )
There is a pair of adjoint (∞,1)-functors
with left adjoint to the canonical functor given by precomposition with .
Definition ( affine -scheme, StSp 2.3.9)
Write for the (∞,1)-functor
A -structured (∞,1)-topos in the image of this functor is an affine -scheme.
Definition (geometric scheme, StSp 2.3.9)
Let be a geometry (for structured (∞,1)-toposes).
A -structured (∞,1)-topos is a -scheme if
- there exists a collection
the cover in that the canonical morphism (with the terminal object of ) is an effective epimorphism;
for every there exists an equivalence
of structured -toposes for some (in the (∞,1)-category of pro-objects of ).
Definition (pregeometric scheme, StSp, 3.4.6)
For a pregeometry, a -structured (infinity,1)-topos is a -scheme if it is a -scheme for the geometric envelope of .
This means that for the geometric envelope and for the -structure on such that , we have that is a -scheme.
Let be a pregeometry (for structured (∞,1)-toposes) and let be an inclusion into an enveloping geometry (for structured (∞,1)-toposes).
We think of the objects of as the smooth test spaces – for instance the cartesian products of some affine line with itsef – and of the objects of as affine test spaces that may have singular points where they are not smooth.
The idea is that a smooth -scheme is a -structured space that is locally not only equivalent to objects in , but even to the very nice – “smooth” – objects in .
Definition ( smooth -scheme, StSp 3.5.6)
With an envelope fixed, a -scheme is called smooth if there the affine schemes appearing in its definition may be chosen with in the image of the includion .
See the discussion at derived scheme for how ordinary schemes are special cases of generalized schemes.
Ordinary Deligne-Mumford stacks
See the discussion at derived Deligne-Mumford stack for how ordinary Deligne-Mumford stacks are special cases of derived Deligne-Mumford stacks.
Definition (derived scheme, Structured Spaces, 4.2.8)
Let be a commutative ring. Recall the pregoemtry .
A derived scheme over is a -scheme.
Derived smooth manifolds
Derived Deligne-Mumford stacks
Derived schemes with -ring valued structure sheaves
The above derived schemes have structure sheaves with values in simplicial commutative rings. There is also a notion of derived scheme whose structure sheaf takes values in E-infinity rings. The theory of these is to be described in full detail in
An indication of some details is in
See at E-∞ scheme and E-∞ geometry.
Generalized schemes are definition 2.3.9 of
The definition of affine -schemes (absolute spectra) is in section 2.2.