nLab coherent (infinity,1)-topos

Idea

The concept of coherent $(\infty,1)$-topos is a notion of compact topos in the context of (∞,1)-topos theory (Lurie VII, def. 3.1).

Definitions

Definition

An (∞,1)-topos $\mathbf{H}$ is called quasi-compact if, for every effective epimorphism

$$\coprod_{i\in I} U_i\to *$$

there exists a finite subset $J\subset I$ such that $\coprod_{i\in J} U_i\to *$ is an effective epimorphism. An object $X\in\mathbf{H}$ is called quasi-compact if the slice (∞,1)-topos $\mathbf{H}_{/X}$ is quasi-compact.

We then define $n$-coherence by induction on $n$.

Definition

Let $\mathbf{H}$ be an (∞,1)-topos. We say that $\mathbf{H}$ is 0-coherent if it is quasi-compact. If $n\geq 1$, we say that $\mathbf{H}$ is n-coherent if

1. it is locally $(n-1)$-coherent, i.e., for every $X\in\mathbf{H}$ there exists an effective epimorphism $\coprod_{i\in I} U_i\to X$ such that each $U_i$ is (n-1)-coherent;
2. the sub-(∞,1)-category of (n-1)-coherent objects in $\mathbf{H}$ is closed under finite products.

We say that $\mathbf{H}$ is coherent if it is $n$-coherent for every $n\geq 0$, and locally coherent if for every $X\in\mathbf{H}$ there exists an effective epimorphism $\coprod_{i\in I} U_i\to X$ such that each $U_i$ is coherent.

(Lurie SpecSch, def. 3.1, def. 3.12)

Remark

This terminology differs from the one in SGA4: a topos is a coherent topos in the sense of SGA4 if and only if it is 2-coherent according to the above definition.

Definition

An object $X \in \mathcal{X}$ in an (∞,1)-topos is a n-coherent object if the slice (∞,1)-topos $\mathcal{X}_{/X}$ is $n$-coherent according to def. .

Properties

Commutativity with filtered colimits

Notice that a compact object in an (∞,1)-category is one that distributes over filtered (∞,1)-colimits.

In an $n$-coherent $\infty$-topos the global section geometric morphism (given by homming out of the terminal object) preserves filtered (∞,1)-colimits of (n-1)-truncated objects.

In terms of sites

An (∞,1)-site is finitary if every covering sieve is generated by a finite family of morphisms. If $C$ is a finitary (∞,1)-site with finite (∞,1)-limits, then the (∞,1)-topos of (∞,1)-sheaves on $C$ is coherent and locally coherent.

Deligne-Lurie completeness theorem

The following generalizes the Deligne completeness theorem from topos theory to (∞,1)-topos theory.

Theorem

Deligne-Lurie completeness theorem

An hypercomplete (∞,1)-topos which is locally coherent has enough points.

(Lurie SpecSchm, theorem 4.1).

Examples

Example

∞Grpd is coherent and locally coherent. An object $X$, hence an ∞-groupoid, is an n-coherent object if all its homotopy groups in degree $k \leq n$ are finite. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.

(Lurie SpecSchm, example 3.13)

Example

Let $X$ be a scheme and let $Sh_\infty(X_{Zar})$ be the (∞,1)-topos of (∞,1)-sheaves on the small Zariski site of $X$. Then the following assertions are equivalent:

1. $Sh_\infty(X_{Zar})$ is coherent;
2. $Sh_\infty(X_{Zar})$ is 1-coherent;
3. $X$ is quasi-compact and quasi-separated.

Example

A spectral scheme or spectral Deligne-Mumford stack, regarded as a structured (∞,1)-topos is locally coherent.

(Lurie QCoh, cor. 1.4.3)

References

• Jacob Lurie, section 3 of Spectral Schemes

• Jacob Lurie, section 2.3 of Rational and p-adic Homotopy Theory

• Jacob Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems