derived smooth geometry
as described at function algebras on ∞-stacks Here is an -algebra of functions on .
This entry describes for certain algebraic stacks an analog of this situation where the 1-algebras are replaced by 2-algebras in the form of commutative algebra objects in the 2-category of abelian categories: abelian symmetric monoidal categories, and where the function algebras are replaced with category of quasicoherent sheaves.
The replacement of the 1-algebra by the 2-algebra is the starting point for what is called derived noncommutative geometry.
All toposes that we consider are Grothendieck toposes. A ringed topos is a topos equipped with a ring object – a sheaf of rings – called the structure sheaf – on whatever site is the category of sheaves on. We write for the category of modules in (sheaves of modules) over .
We write for the category of ringed toposes.
is faithfully flat, there exists morphisms in and factorizations such that
is an epimorphism.
This is (Lurie, remark 4.4).
An abelian tensor category (for the purposes of the present discusission) is a symmetric monoidal category such that
is an abelian category;
A complete abelian tensor category is an abelian tensor category such that
it satisfies the axiom AB5 at additive and abelian categories;
commutes with all small colimits.
(equivalently, we have a closed monoidal category).
An abelian tensor category is called tame if for any short exact sequence
with a flat object (such that is an exact functor) and any also the induced sequence
This appears as (Lurie, def. 5.2) together with the paragraph below remark 5.3.
For two complete abelian tensor categories write
preserve flat objects and short exact sequences whose last object is flat.
This appears as (Lurie, def 5.9) together with the following remarks.
Let be a ringed topos. Then
This is (Lurie, example 5.7).
For an algebraic stack, write
for its category quasicoherent sheaves.
This is a complete abelian tensor category
This appears as (Lurie, lemma 3.9).
A geometric stack is
The geometricity condition on an algebraic stack implies that there are “enough” quasicoherent sheaves on it, as formalized by the following statement.
If is a geometric stack then the bounded-below derived category of quasicoherent sheaves on is naturally equivalent to the full subcategory of the left-bounded derived category of smooth-etale -modules whose chain cohomology sheaves are quasicoherent.
This is (Lurie, theorem 3.8).
Let be a geometric stack.
Then for every ring there is an equivalence of categories
hence (by the 2-Yoneda lemma)
from geometric stacks to tame complete abelian tensor categories.
This statement justifies thinking of as being the “2-algebra” of functions on . This perspective is the basis for derived noncommutative geometry.
The above material is taken from
The generalization to geometric stacks in the context of Spectral Schemes is in
Related discussion is in