Fractured topoi | Higher Topos Theory notes

7.3. Fractured topoi

A geometry produces two closely related sheaf topoi: the topos of all sheaves on the affine test objects, and the topos obtained by restricting to admissible morphisms. Lurie's notion of fractured topos abstracts the structure retained by such a pair. In the Zariski case, this is the structure behind the four equivalent descriptions encountered above: corporeal objects, admissible morphisms, factorization systems, and local morphisms of structured topoi.

Definition 7.12.

A fractured topos is a topos $$E$$ equipped with a functor $j_!\colon E^{\corp} \to E$ satisfying the following properties:

The functor $$j_!$$ is faithful and induces a fully faithful functor on groupoid cores.
The category $E^{\corp}$ admits fiber products and $$j_!$$ preserves them.
The functor $$j_!$$ admits a right adjoint $$j^*$$ which is conservative and preserves colimits.
For every morphism $U \to V$ in $E^{\corp}$ , the commutative square

is a pullback square.

We refer to objects of $E^{\corp}$ , and to their images in $$E$$ , as corporeal objects. The functor $$j_!$$ is generally not fully faithful. The fractured topos is called complete if every retract in $$E$$ of a corporeal object is corporeal.

Proposition 7.13.

Let $(E,E^{\corp})$ be a fractured topos. Then:

The category $E^{\corp}$ is a topos.
For every $X\in E^{\corp}$ , the functor
$j_!\colon E^{\corp}_{/X}\longrightarrow E_{/j_!X}$
is fully faithful.
The topos $$E$$ is generated under colimits by corporeal objects.
The functor $$j^*$$ admits a right adjoint $j_*\colon E^{\corp}\to E$ , and the adjunction $j^*\dashv j_*$ is a morphism of topoi.

Proof

See [Lurie 2018, Propositions 20.1.3.1 and 20.1.3.3, Corollary 20.1.3.4, and Remark 20.1.3.6].

We will now discuss how to construct fractured topoi.

7.3.1. Fractures from geometries

Construction 7.14.

Let $(C,C^{\ad},\tau)$ be a geometric site. The category $C^{\ad}$ inherits a topology $\tau^{\ad}$ : a sieve in $C^{\ad}_{/X}$ is covering if its image generates a $\tau$ -covering sieve in $C_{/X}$ . Restriction along $j\colon C^{\ad}\hookrightarrow C$ defines a functor

j^*\colon \Shv_{\tau}(C)\longrightarrow \Shv_{\tau^{\ad}}(C^{\ad}).

Its left adjoint $$j_!$$ is obtained by left Kan extension followed by sheafification.

Theorem 7.15.

The functor

j_!\colon \Shv_{\tau^{\ad}}(C^{\ad})\longrightarrow \Shv_{\tau}(C)

exhibits $\Shv_{\tau}(C)$ as a complete fractured topos.

Proof

See [Lurie 2018, Theorem 20.6.3.4]; completeness follows from the associated geometric admissibility structure, by [Lurie 2018, Theorem 20.3.4.4].

Example 7.16.

Every geometry listed in Section 7.2 gives a fractured topos by this construction. For instance, the classical Zariski geometry gives the pair

\Shv_{\Zar}((\CRing^{\mathrm{fp},\ad})\catop) \longrightarrow \Shv_{\Zar}((\CRing^{\mathrm{fp}})\catop),

which is the fractured topos underlying the topos-theoretic description of ordinary schemes above.

Definition 7.17.

Let $(E,E^{\corp})$ be a fractured topos. A morphism $f\colon Y\to X$ in $$E$$ is admissible if, for every map $j_!U\to X$ with $U\in E^{\corp}$ , the pullback

Y\times_Xj_!U\longrightarrow j_!U

belongs to the image of $E^{\corp}_{/U}\to E_{/j_!U}$ . This determines a wide subcategory $E^{\ad}\subseteq E$ .

Definition 7.18.

Let $$E$$ be a topos. An admissibility structure $E^{\ad}$ is called local if the following conditions are satisfied:

The class $E^{\ad}$ is a local class of maps.
For every $X \in E$ , the slice $E^{\ad}_{/X} := (E^{\ad})_{/X}$ is presentable, and the inclusion $E^{\ad}_{/X} \to E_{/X}$ preserves colimits. In particular, it admits a right adjoint $\rho_X\colon E_{/X} \to E^{\ad}_{/X}$ .

Given a local admissibility structure on $$E$$ , we say that an object $X \in E$ is corporeal if the functor $\rho_X\colon E_{/X} \to E^{\ad}_{/X}$ preserves colimits. We denote by

E^{\corp} \subseteq E^{\ad}

the full subcategory spanned by the corporeal objects. We say that $E^{\ad}$ is geometric if, moreover, $$E$$ is generated under colimits by corporeal objects.

Theorem 7.19.

Let $$E$$ be a topos. The assignments $(E,E^{\corp}) \mapsto (E,E^{\ad})$ and $(E,E^{\ad}) \mapsto (E,E^{\corp})$ determine an equivalence of posets

\{\text{Complete fractured structures on $E$}\} \simeq \{\text{Geometric admissibility structures on $E$}\}.

Proof

This is [Lurie 2018, Theorem 20.3.4.4 and Remark 20.3.4.6]. The completeness hypothesis is essential: without it, the admissible morphisms recover the closure of the corporeal objects under retracts, rather than necessarily recovering the original fracture.

7.3.2. Local maps

Consider a fractured topos $(E,E^{\corp})$ . Consider two morphisms of topoi $f,g\colon T \to E$ , and let $\alpha\colon f^* \to g^*$ be a natural transformation. We say that $\alpha$ is local if for every morphism $h\colon U \to V$ in $E^{\corp}$ , the following square is a pullback square:

This determines a collection of maps in $\Geom(T,E)$ which is the right class of a factorization system on $\Geom(T,E)$ . We denote by

(\Topos_{//E})^{\loc} \subseteq \Topos_{//E}

the wide subcategory consisting of those morphisms corresponding to maps $S \to T$ with a local natural transformation $\alpha$ of morphisms $S \to E$ .

Example 7.20.

For $E = \PSh(\CRing^{\mathrm{fp},\op})$ , the lax slice $\Topos_{//E}$ is the category of ringed topoi. For $E = \Shv_{\Zar}(\CRing^{\mathrm{fp},\op})$ , the lax slice $\Topos_{//E}$ is the full subcategory of the previous one spanned by the locally ringed topoi. Here, the fracture $E^{\corp} = \Shv_{\Zar}(\CRing^{\mathrm{fp}, \ad, \op}) \to E$ determines the subcategory $(\Topos_{//E})^{\loc}$ consisting of the local morphisms of locally ringed topoi.

We now explain how to recover the notion of schemes from the notion of local morphisms.

Theorem 7.21.

Let $(E,E^{\corp})$ be a complete fractured topos. Consider the functor

E \to \Fun(\Topos^{\loc}_{//E}, \An), \qquad X \mapsto \Gamma^X : (f\colon T \to E) \mapsto \Gamma(f^*(X)).

If $$X$$ is corporeal, then $\Gamma^X$ is representable by the pair $(E^{\ad}_{/X}, E^{\ad}_{/X} \hookrightarrow E_{/X} \to E)$ .
The resulting functor
$E^{\corp} \xrightarrow{\mathrm{Re}} \Topos^{\loc}_{//E}$
is fully faithful.
A morphism $f\colon Y \to X$ in $$E$$ is admissible if and only if, after every base change $f'\colon Y'\to X'$ for which $\Gamma^{X'}$ is represented by an object of $\Topos^{\loc}_{//E}$ , the functor $\Gamma^{Y'}$ is represented by an object étale over it.

Combining these statements, it follows that the subcategory $\Topos^{\loc}_{//E}\subseteq \Topos_{//E}$ determines the fracture structure on $$E$$ .

Proof

These assertions summarize the reconstruction results of [Lurie 2018, Chapter 20]. The third assertion uses the usual identification of étale objects over a topos with objects of that topos.

7.3.3. Schemes

In Section 7.1, we provided a topos-theoretic perspective on the classical notion of a scheme. We may extend this definition to an arbitrary geometric site $\Gg = (C,C^{\ad},\tau)$ .

Construction 7.22.

Consider the fractured topos $E = \Shv_{\tau}(C)$ , $E^{\corp} := \Shv_{\tau}(C^{\ad})$ . By Theorem 7.21, we obtain a functor

\Gamma\colon \Topos^{\loc}_{//E} \to \Pt(\PSh(C))\catop \simeq \Pro(C),

An object $f\colon T\to E$ is sent to the pro-object classified by the left exact functor

C \hookrightarrow \PSh(C) \twoheadrightarrow \Shv_{\tau}(C) = E \xrightarrow{f^*} T \xrightarrow{\Gamma} \An.

The functor $\Gamma$ admits a right adjoint

\Spec_{\Gg}\colon \Pro(C) \to \Topos^{\loc}_{//E}

called the spectrum functor. If $\tau$ is subcanonical, then $\Spec_{\Gg}$ is fully faithful. See [Lurie 2011, Chapter 2] for this structured-topos formulation of the spectrum construction.

Definition 7.23.

We define the subcategory

\Sch_{\Gg} \quad \subseteq \quad \Topos^{\loc}_{//E}

of $\Gg$ -schemes as the full subcategory on those objects which are locally in the image of $\Spec_{\Gg}$ . Concretely, a $\Gg$ -scheme admits a jointly surjective family of étale morphisms from affine objects $\Spec_{\Gg}(A)$ with $A\in\Pro(C)$ .

Question 7.24.

Does this category depend only on the fractured topos $(E,E^{\corp})$ , or is the choice of geometric site important?

References

Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.
Jacob Lurie. Derived algebraic geometry V: Structured spaces. 2011.