Contents

topos theory

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Idea

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

$(\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_{(\infty,1)}(C)$

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.

## Setup

### Ringed toposes

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.

###### Definition

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

$\mathcal{O}_S(U) \to \prod_i R_i$

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

$\coprod_i E_i \to E$

is an epimorphism.

###### Proposition

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).

###### Example
• The little étale topos $Sh(X_{et})$ of a Deligne-Mumford stack $X$ is locally ringed with respect to the étale topology.

### Abelian tensor categories

###### Definition

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

An abelian tensor category is called tame if for any short exact sequence

$0 \to M'\to M \to M''\to 0$

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

$0 \to M'\otimes N \to M\otimes N \to M''\otimes N \to 0$

is exact.

This appears as (Lurie, def. 5.2) together with the paragraph below remark 5.3.

###### Definition

For $C,D$ two complete abelian tensor categories write

$Func_\otimes(C,D) \subset Func(C,D)$

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

$TCAbTens$

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.

###### Example

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

$\mathcal{O}_S Mod$

(the category of sheaves of $\mathcal{O}_S$-modules) is a tame complete abelian tensor category.

This is (Lurie, example 5.7).

###### Example

For $X$ an algebraic stack, write

$QC(X)$

for its category quasicoherent sheaves.

This is a complete abelian tensor category

###### Lemma

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

$QC(X) \simeq Ind(Coh(X)) \,.$

This appears as (Lurie, lemma 3.9).

### Geometric stacks

###### Definition
• 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 $X \to X \times X$.

###### Example

The geometricity condition on an algebraic stack implies that there are “enough” quasicoherent sheaves on it, as formalized by the following statement.

###### Theorem

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).

## Tannaka duality for geometric stacks

###### Theorem

Let $X$ be a geometric stack.

Then for every ring $A$ there is an equivalence of categories

$RngdTopos(Sh((Spec A)_{et}),Sh(X_{et})) \simeq Hom_\otimes(QC(X), A Mod)$

hence (by the 2-Yoneda lemma)

$X(Spec A) \simeq Hom_\otimes(QC(X), QC(Spec A)) \,.$

More generally, for $(S, \mathcal{O}_S)$ any etale-locally ringed topos, we have

$RngdTopos(S,Sh(X_{et})) \simeq Hom_\otimes(QC(X), \mathcal{O}_S Mod) \,.$

This is (Lurie, theorem 5.11) in view of (Lurie, remark 4.5).

###### Remark

It follows that forming quasicoherent sheaves constitutes a full and faithful (2,1)-functor

$QC : GeomStacks \to TCAbTens^{op}$

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