Artin's representability theorem gives conditions for a functor from commutative rings to sets (or groupoids) to be representable by an algebraic space (stack). The conditions are largely expressed in terms of the formal deformation theory. One key condition is the existence of an obstruction theory.

Lurie's representability theorem is a similar looking statement in "spectral" or "derived" algebraic geometry. It gives similar looking conditions for a functor from $E_\infty$ rings to spaces to be representable by a spectral scheme or stack. One notable difference is that the condition on the existence of an obstruction theory (which seems to be auxilliary data) is replaced by a condition on the cotangent complex

Sometimes I hear that Lurie's theorem is a generalization of Artin's theorem.

Does the statement of Lurie's representability theorem imply the statement of Artin's representability theorem?

Let me clarify the question a bit. Lurie's theorem takes as input vastly more data than Artin's theorem: whereas Artin asks for a functor from commutative rings to sets, Lurie asks for a functor defined on spectral commutative rings. So, to get Artin from Lurie, one must know how to extend a functor from commutative rings to spectral commutative rings.

There is one remark in Lurie's article that hints at the answer to this: "In the setting of Artin’s original theorem, a deformation and obstruction theory are auxiliary constructs which are not uniquely determined by the functor F. The meaning of these conditions are clarified by working in the spectral setting: they are related to the problem of extending the functor F to $E_\infty$-rings which are nondiscrete."

So a more precise version of the above question is:

How does one use a deformation/obstruction theory to extend a functor from commutative rings to sets (satisfying Artin's axioms) to a functor from $E_\infty$ rings to spaces (satisfying Lurie's axioms)?

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    $\begingroup$ There is no general procedure to extend a moduli problem to derived/spectral commutative rings (except in a trivial way). However in practice there is usually a natural extension. Given such an extension, the cotangent complex determines a canonical deformation/obstruction theory, so that this is no longer auxiliary data. But I don't believe it to be the case that a fixing a deformation/obstruction theory determines a unique derived extension of your functor. $\endgroup$ – Riza Hawkeye Nov 28 at 22:17
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    $\begingroup$ Perhaps I should also clarify that it's not really an issue to construct derived versions of moduli spaces. Pretty much any interesting moduli problem that you can think of will tend to have a natural derived version with a more or less obvious definition. Which is why Lurie's representability theorem is so useful; once the functor is written down, it's typically straightforward to check Lurie's criteria for representability. $\endgroup$ – Riza Hawkeye Nov 28 at 22:21
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    $\begingroup$ Perhaps my second question would have been more precisely stated as: "(how) does a deformation/obstruction theory guarantee the existence of an extension of a functor from..." -- the uniqueness of said extension is immaterial to deducing Artin's theorem from Lurie's. $\endgroup$ – Vivek Shende Nov 29 at 2:42
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    $\begingroup$ Existence of deformation/obstruction theory, or even of a perfect obstruction theory, does not guarantee existence of a derived enhancement. There are obstructions described in arxiv.org/pdf/1005.3945v4.pdf . $\endgroup$ – Jon Pridham Nov 29 at 7:18
  • $\begingroup$ Thanks @JonPridham — I guess that’s as good as an answer as I can hope for. Does it mean Lurie representability does not imply Artin representability? Is there a common generalization, e.g. I ask the functor to only be defined on n-truncated things and search for a representing n-truncated stack? $\endgroup$ – Vivek Shende Nov 29 at 19:42

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