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symmetric monoidal (∞,1)-category of spectra
The generalization of the Artin representability theorem from algebraic geometry to spectral algebraic geometry.
Write for the (∞,1)-category of connective E-∞ rings, and ∞Grpd for that of ∞-groupoids.
Necessary and sufficient conditions for an (∞,1)-presheaf
over some , 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 :
For every discrete commutative ring, is n-truncated.
is an ∞-stack for the étale (∞,1)-site.
is nilcomplete, integrable, and an infinitesimally cohesive (∞,1)-presheaf on E-∞ rings.
admits a connective cotangent complex.
the natural transformation to is locally almost of finite presentation.
The condition that 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.