(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The concept of coherent -topos is a notion of compact topos in the context of (∞,1)-topos theory (Lurie VII, def. 3.1).
An (∞,1)-topos is called quasi-compact if, for every effective epimorphism
there exists a finite subset such that is an effective epimorphism. An object is called quasi-compact if the slice (∞,1)-topos is quasi-compact.
We then define -coherence by induction on .
Let be an (∞,1)-topos. We say that is 0-coherent if it is quasi-compact. If , we say that is n-coherent if
We say that is coherent if it is -coherent for every , and locally coherent if for every there exists an effective epimorphism such that each 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 in an (∞,1)-topos is a n-coherent object if the slice (∞,1)-topos is -coherent according to def. .
Notice that a compact object in an (∞,1)-category is one that distributes over filtered (∞,1)-colimits.
In an -coherent -topos the global section geometric morphism (given by homming out of the terminal object) preserves filtered (∞,1)-colimits of (n-1)-truncated objects.
An (∞,1)-site is finitary if every covering sieve is generated by a finite family of morphisms. If is a finitary (∞,1)-site with finite (∞,1)-limits, then the (∞,1)-topos of (∞,1)-sheaves on is coherent and locally coherent.
The following generalizes the Deligne completeness theorem from topos theory to (∞,1)-topos theory.
Deligne-Lurie completeness theorem
An hypercomplete (∞,1)-topos which is locally coherent has enough points.
(Lurie SpecSchm, theorem 4.1).
∞Grpd is coherent and locally coherent. An object , hence an ∞-groupoid, is an n-coherent object if all its homotopy groups in degree are finite. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.
(Lurie SpecSchm, example 3.13)
Let be a scheme and let be the (∞,1)-topos of (∞,1)-sheaves on the small Zariski site of . Then the following assertions are equivalent:
A spectral scheme or spectral Deligne-Mumford stack, regarded as a structured (∞,1)-topos is locally coherent.
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
Last revised on June 23, 2019 at 21:33:24. See the history of this page for a list of all contributions to it.