topos theory

# Contents

## Idea

Under mild conditions, a given site $C\subset T{\mathrm{Alg}}^{\mathrm{op}}$ of formal duals of algebras over an algebraic theory admits Isbell duality exhibited by an adjunction

$\left(𝒪⊣\mathrm{Spec}\right):\left(T{\mathrm{Alg}}^{\Delta }{\right)}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{←}}{\underset{\mathrm{Spec}}{\to }}{\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)$(\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 $𝒪\left(X\right)$ is an $\left(\infty ,1\right)$-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 $𝒪\left(X\right)$ are replaced with category $\mathrm{QC}\left(X\right)$ of quasicoherent sheaves.

The replacement of the 1-algebra $𝒪\left(X\right)$ by the 2-algebra $\mathrm{QC}\left(X\right)$ 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 $\left(S,{𝒪}_{S}\right)$ is a topos $S$ equipped with a ring object ${𝒪}_{S}$ – a sheaf of rings – called the structure sheaf – on whatever site $S$ is the category of sheaves on. We write ${𝒪}_{S}\mathrm{Mod}$ for the category of modules in $S$ (sheaves of modules) over ${𝒪}_{S}$.

We write $\mathrm{RngdTopos}$ for the category of ringed toposes.

For $X$ a scheme or more generally an algebraic stack, write $\mathrm{Sh}\left({X}_{\mathrm{et}}\right)$ for its little etale topos.

###### Definition

A ringed topos $\left(S,{𝒪}_{S}\right)$ is a locally ringed topos with respect to the étale topology if for every object $U\in S$ and every family $\left\{\mathrm{Spec}{R}_{i}\to \mathrm{Spec}{𝒪}_{S}\left(U\right)\right\}$ of étale morphisms such that

${𝒪}_{S}\left(U\right)\to \prod _{i}{R}_{i}$\mathcal{O}_S(U) \to \prod_i R_i

is faithfully flat?, there exists morphisms ${E}_{i}\to E$ in $S$ and factorizations ${𝒪}_{S}\left(U\right)\to {R}_{i}\to {𝒪}_{S}\left({E}_{i}\right)$ such that

$\coprod _{i}{E}_{i}\to E$\coprod_i E_i \to E

is an epimorphism.

###### Proposition

If $S$ has enough points then $\left(S,{𝒪}_{S}\right)$ is local for the étale topology precisely if the stalk ${𝒪}_{S}\left(x\right)$ at every point $x:\mathrm{Set}\to S$ is a strictly Henselian local ring.

This is (Lurie, remark 4.4).

###### Example
• The little étale topos $\mathrm{Sh}\left({X}_{\mathrm{et}}\right)$ 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 $\left(C,\otimes \right)$ such that

• $C$ is an abelian category;

• for every $x\in C$ the functor $\left(-\right)\otimes x:C\to C$ 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\prime \to M\to M″\to 0$0 \to M'\to M \to M''\to 0

with $M″$ a flat object (such that $x↦x\otimes M″$ is an exact functor) and any $N\in C$ also the induced sequence

$0\to M\prime \otimes N\to M\otimes N\to M″\otimes N\to 0$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

${\mathrm{Func}}_{\otimes }\left(C,D\right)\subset \mathrm{Func}\left(C,D\right)$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

$\mathrm{TCAbTens}$TCAbTens

for the (strict) (2,1)-category of tame complete abelian tensor categories with hom-groupoids given by this ${\mathrm{Func}}_{\otimes }$.

This appears as (Lurie, def 5.9) together with the following remarks.

###### Example

For $k$ a ring, write $k\mathrm{Mod}$ for its abelian symmetric monoidal category of modules

Let $\left(S,{𝒪}_{S}\right)$ be a ringed topos. Then

${𝒪}_{S}\mathrm{Mod}$\mathcal{O}_S Mod

(the category of sheaves of ${𝒪}_{S}$-modules) is a tame complete abelian tensor category.

This is (Lurie, example 5.7).

###### Example

For $X$ an algebraic stack, write

$\mathrm{QC}\left(X\right)$QC(X)

for its category quasicoherent sheaves.

This is a complete abelian tensor category

###### Lemma

If $X$ is a Noetherian geometric stack, then $\mathrm{QC}\left(X\right)$ is the category of ind-objects of its full subcategory $\mathrm{Coh}\left(X\right)\subset \mathrm{QC}\left(X\right)$ of coherent sheaves

$\mathrm{QC}\left(X\right)\simeq \mathrm{Ind}\left(\mathrm{Coh}\left(X\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$QC(X) \simeq Ind(Coh(X)) \,.

This appears as (Lurie, lemma 3.9).

### Geometric stacks

###### Definition
• an algebraic stack $X$ over $\mathrm{Spec}ℤ$

• that is quasi-compact, in particular there is an epimorphism $\mathrm{Spec}A\to X$;

• with affine and representable diagonal $X\to X×X$.

###### 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 ${𝒪}_{X}$-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

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

hence (by the 2-Yoneda lemma)

$X\left(\mathrm{Spec}A\right)\simeq {\mathrm{Hom}}_{\otimes }\left(\mathrm{QC}\left(X\right),\mathrm{QC}\left(\mathrm{Spec}A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$X(Spec A) \simeq Hom_\otimes(QC(X), QC(Spec A)) \,.

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

$\mathrm{RngdTopos}\left(S,\mathrm{Sh}\left({X}_{\mathrm{et}}\right)\right)\simeq {\mathrm{Hom}}_{\otimes }\left(\mathrm{QC}\left(X\right),{𝒪}_{S}\mathrm{Mod}\right)\phantom{\rule{thinmathspace}{0ex}}.$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

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

from geometric stacks to tame complete abelian tensor categories.

This statement justifies thinking of $\mathrm{QC}\left(X\right)$ as being the “2-algebra” of functions on $X$. This perspective is the basis for derived noncommutative geometry.

## References

The above material is taken from

The generalization to geometric stacks in the context of Spectral Schemes is in

Related discussion is in

Revised on February 4, 2013 18:21:26 by Urs Schreiber (82.113.99.102)