higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
symmetric monoidal (∞,1)-category of spectra
The generalization of the Artin representability theorem from algebraic geometry to spectral algebraic geometry.
Write $CAlg^{cn}$ for the (∞,1)-category of connective E-∞ rings, and ∞Grpd for that of ∞-groupoids.
Necessary and sufficient conditions for an (∞,1)-presheaf
over some $Spec R$, on the opposite (∞,1)-category of connective E-∞ rings to be represented by a spectral Deligne-Mumford n-stack locally of almost finite presentation over $R$:
For every discrete commutative ring, $\mathcal{F}(A)$ is n-truncated.
$\mathcal{F}$ is an ∞-stack for the étale (∞,1)-site.
$\mathcal{F}$ is nilcomplete, integrable, and an infinitesimally cohesive (∞,1)-presheaf on E-∞ rings.
$\mathcal{F}$ admits a connective cotangent complex.
the natural transformation to $Spec R$ is locally almost of finite presentation.
The condition that $\mathcal{F}$ be infinitesimally cohesive implies that the Lie differentiation around any point, given by restriction to local Artin rings (formal duals of infinitesimally thickened points), is a formal moduli problem, hence equivalently an L-∞ algebra.
The motivating example of the Artin-Lurie representability theorem is the re-proof of the Goerss-Hopkins-Miller theorem. See there for more.
Last revised on July 29, 2016 at 18:20:03. See the history of this page for a list of all contributions to it.