higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Under mild conditions, a given site $C \subset T Alg^{op}$ of formal duals of algebras over an algebraic theory admits Isbell duality exhibited by an adjunction
as described at function algebras on ∞-stacks Here $\mathcal{O}(X)$ is an $(\infty,1)$-algebra of functions on $X$.
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 $\mathcal{O}(X)$ are replaced with category $QC(X)$ of quasicoherent sheaves.
The replacement of the 1-algebra $\mathcal{O}(X)$ by the 2-algebra $QC(X)$ is the starting point for what is called derived noncommutative geometry.
All toposes that we consider are Grothendieck toposes. A ringed topos $(S, \mathcal{O}_S)$ is a topos $S$ equipped with a ring object $\mathcal{O}_S$ – a sheaf of rings – called the structure sheaf – on whatever site $S$ is the category of sheaves on. We write $\mathcal{O}_S Mod$ for the category of modules in $S$ (sheaves of modules) over $\mathcal{O}_S$.
We write $RngdTopos$ for the category of ringed toposes.
For $X$ a scheme or more generally an algebraic stack, write $Sh(X_{et})$ for its little etale topos.
A ringed topos $(S,\mathcal{O}_S)$ is a locally ringed topos with respect to the étale topology if for every object $U \in S$ and every family $\{Spec R_i \to Spec \mathcal{O}_S(U)\}$ of étale morphisms such that
is faithfully flat, there exists morphisms $E_i \to E$ in $S$ and factorizations $\mathcal{O}_S(U) \to R_i \to \mathcal{O}_S(E_i)$ such that
is an epimorphism.
If $S$ has enough points then $(S, \mathcal{O}_S)$ is local for the étale topology precisely if the stalk $\mathcal{O}_S(x)$ at every point $x : Set \to S$ is a strictly Henselian local ring.
This is (Lurie, remark 4.4).
An abelian tensor category (for the purposes of the present discusission) is a symmetric monoidal category $(C, \otimes)$ such that
$C$ is an abelian category;
for every $x \in C$ the functor $(-) \otimes x : C\to C$ is additive and right-exact: it commutes with finite colimits.
A complete abelian tensor category is an abelian tensor category such that
it satisfies the axiom AB5 at additive and abelian categories;
$(-) \otimes x$ 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 $M''$ a flat object (such that $x \mapsto x \otimes M''$ is an exact functor) and any $N \in C$ also the induced sequence
is exact.
This appears as (Lurie, def. 5.2) together with the paragraph below remark 5.3.
For $C,D$ two complete abelian tensor categories write
for the core of the subcategory of the functor category on those functors that
commute with all small colimits (which implies they are additive and right exact)
preserve flat objects and short exact sequences whose last object is flat.
Write
for the (strict) (2,1)-category of tame complete abelian tensor categories with hom-groupoids given by this $Func_\otimes$.
This appears as (Lurie, def 5.9) together with the following remarks.
For $k$ a ring, write $k Mod$ for its abelian symmetric monoidal category of modules
Let $(S,\mathcal{O}_S)$ be a ringed topos. Then
(the category of sheaves of $\mathcal{O}_S$-modules) is a tame complete abelian tensor category.
This is (Lurie, example 5.7).
For $X$ an algebraic stack, write
for its category quasicoherent sheaves.
This is a complete abelian tensor category
If $X$ is a Noetherian geometric stack, then $QC(X)$ is the category of ind-objects of its full subcategory $Coh(X) \subset QC(X)$ of coherent sheaves
This appears as (Lurie, lemma 3.9).
A geometric stack is
an algebraic stack $X$ over $Spec \mathbb{Z}$
that is quasi-compact, in particular there is an epimorphism $Spec A \to X$;
with affine and representable diagonal $X \to X \times X$.
A quasicompact separated scheme is a geometric stack.
The classifying stack of a smooth affine group scheme is a geometric stack.
The geometricity condition on an algebraic stack implies that there are “enough” quasicoherent sheaves on it, as formalized by the following statement.
If $X$ is a geometric stack then the bounded-below derived category of quasicoherent sheaves on $X$ is naturally equivalent to the full subcategory of the left-bounded derived category of smooth-etale $\mathcal{O}_X$-modules whose chain cohomology sheaves are quasicoherent.
This is (Lurie, theorem 3.8).
Let $X$ be a geometric stack.
Then for every ring $A$ there is an equivalence of categories
hence (by the 2-Yoneda lemma)
More generally, for $(S, \mathcal{O}_S)$ any etale-locally ringed topos, we have
This is (Lurie, theorem 5.11) in view of (Lurie, remark 4.5).
It follows that forming quasicoherent sheaves constitutes a full and faithful (2,1)-functor
from geometric stacks to tame complete abelian tensor categories.
This statement justifies thinking of $QC(X)$ as being the “2-algebra” of functions on $X$. 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