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

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.

Definition

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

Definition

An (∞,1)-functor

$$X \;\colon\; CRing_\infty^{cn}\longrightarrow \infty Grpd$$

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

If at least those fiber products whose kernels are nilpotent ideals are preserved, then $X$ 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

• Jacob Lurie, section 2.1 of Representability theorems