coherent (infinity,1)-topos


(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

Compact objects



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



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

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

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

We then define nn-coherence by induction on nn.


Let H\mathbf{H} be an (∞,1)-topos. We say that H\mathbf{H} is 0-coherent if it is quasi-compact. If n1n\geq 1, we say that H\mathbf{H} is n-coherent if

  1. it is locally (n1)(n-1)-coherent, i.e., for every XHX\in\mathbf{H} there exists an effective epimorphism iIU iX\coprod_{i\in I} U_i\to X such that each U iU_i is (n-1)-coherent;
  2. the sub-(∞,1)-category of (n-1)-coherent objects in H\mathbf{H} is closed under finite products.

We say that H\mathbf{H} is coherent if it is nn-coherent for every n0n\geq 0, and locally coherent if for every XHX\in\mathbf{H} there exists an effective epimorphism iIU iX\coprod_{i\in I} U_i\to X such that each U iU_i is coherent.

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


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.


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


Commutativity with filtered colimits

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

In an nn-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 CC is a finitary (∞,1)-site with finite (∞,1)-limits, then the (∞,1)-topos of (∞,1)-sheaves on CC is coherent and locally coherent.

Deligne-Lurie completeness theorem

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

(Lurie SpecSchm, theorem 4.1).



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

(Lurie SpecSchm, example 3.13)


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

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

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

(Lurie QCoh, cor. 1.4.3)