geometric infinity-stack


Higher geometry

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

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos



A geometric \infty-stack is an ∞-stack over a geometry with function theory (𝒪Spec)(\mathcal{O} \dashv Spec) which is an ∞-groupoid that is degreewise an \infty-stack in the image of SpecSpec.

This generalizes the notion of geometric stack from topos theory to (∞,1)-topos theory.


The text below follows (Toën 00). Needs to be expanded and (Toën-Vezzosi 04, Lurie) needs to be brought into the picture…

We consider the higher geometry encoded by a Lawvere theory TT via Isbell duality. Write TAlgT Alg for the category of algebras over a Lawvere theory and write TAlg ΔT Alg^{\Delta} for the (∞,1)-category of cosimplicial TT-algebras .

Consider a site TCTAlg opT \subset C \subset T Alg^{op} that satisfies the assumptions described at function algebras on ∞-stacks. Then, by the discussion given there, we have a pair of adjoint (∞,1)-functors

(𝒪Spec):(TAlg Δ) opSpec𝒪Sh (C)=:H, (\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_\infty(C) =: \mathbf{H} \,,

where H:=Sh (C)\mathbf{H} := Sh_\infty(C) is the (∞,1)-category of (∞,1)-sheaves over CC, the big topos for the higher geometry over CC.


An object XHX \in \mathbf{H} is called a geometric \infty-stack over CC if there is it is the (∞,1)-colimit

Xlim K X \simeq {\lim_\to} K_\bullet

over a groupoid object K :ΔHK_\bullet : \Delta \to \mathbf{H} in H\mathbf{H} such that

  1. K 0K_0 and K 1K_1 are in the image of Spec:(TAlg Δ) opHSpec : (T Alg^{\Delta})^{op} \to \mathbf{H};

  2. the target map d 1:K 1K 0d_1 : K_1 \to K_0 is (…sufficiently well behaved…)

For TT the theory of commutative associative algebras over a commutative ring kk and CC the fpqc topology this appears as Toën 00, definition 4.1.4.




Geometric \infty-stacks are stable under (∞,1)-pullbacks along morphism in the image of SpecSpec.


Use that in the (∞,1)-topos H\mathbf{H} we have universal colimits and that SpecSpec is right adjoint.

Presentation by Kan-fibrant simplicial objects

A presentation of geometric \infty-stacks, in some generality, by suitably Kan-fibrant simplicial objects is in (Pridham 09). See also at Kan-fibrant simplicial manifold.



The notion of geometric \infty-stack as a weak quotient of affine \infty-stacks is considered in section 4 of

More general theory in the context of derived algebraic geometry is in

and specifically in E-∞ geometry in

Discussion of presentation of geometric \infty-stacks by Kan-fibrant simplicial objects in the site is in

Revised on April 8, 2014 10:05:04 by Urs Schreiber (