Contents

Idea

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

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

Definition

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

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

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

For $T$ the theory of commutative associative algebras over a commutative ring $k$ and $C$ the fpqc topology this appears as Toën, definition 4.1.4.

Properties

Examples

• Every object in the image of $\mathrm{Spec}:T{\mathrm{Alg}}_{\mathrm{\infty }}^{\mathrm{op}}\to H$ is a geometric $\mathrm{\infty }$-stack.

• Over the étale site an algebraic stack that is a geometric stack is also a geometric $\mathrm{\infty }$-stack.

• Every schematic homotopy type is given by a geometric $\mathrm{\infty }$-stack.

Related concepts

• geometric stack

• geometric $\mathrm{\infty }$-stack, function algebras on ∞-stacks

References

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

• Bertrand Toën, Affine stacks (Champs affines) (arXiv:math/0012219)