nLab
cohesive (∞,1)-presheaf on E-∞ rings

Contents

This entry is about a weak representability condition on (∞,1)-presheaves in E-∞ geometry. Intuitively this expresses similar behaviour as discussed at cohesion and infinitesimal cohesion, but the definitions themselves are independent and unrelated and apply in somewhat disjoint contexts.

Context

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Higher geometry

Higher algebra

Contents

Definition

Write CRing cnCRing_\infty^{cn} for the (∞,1)-category of connective E-∞ rings.

Definition

An (∞,1)-functor

X:CRing cnGrpd X \;\colon\; CRing_\infty^{cn}\longrightarrow \infty Grpd

((∞,1)-presheaf on (CRing cn) op(CRing_\infty^{cn})^{op}) is called cohesive (Lurie Rep, def. 2.1.1) if it sends (∞,1)-fiber products of morphisms which are surjective on π 0\pi_0 to (∞,1)-fiber products.

If at least those fiber products whose kernels are nilpotent ideals are preserved, then XX is called infinitesimally cohesive.

(Lurie Rep, def. 2.1.9).

Remark

Infinitesimal cohesion, def. , is (together with the property that the base ring is sent to a contractible space) the defining property that makes such a functor a formal moduli problem, hence equivalently an L-∞ algebra. It is also one of the characteristics of a Deligne-Mumford stack, due to the Artin-Lurie representability theorem, hence it is satisfied by those functors which are infinitesimal approximations to geometric infinity-stacks.

References

Last revised on January 16, 2016 at 08:14:15. See the history of this page for a list of all contributions to it.