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Moduli problems for ring spectra. (English) Zbl 1244.55007

Bhatia, Rajendra (ed.) et al., Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency (ISBN 978-981-4324-32-8/hbk; 978-81-85931-08-3/hbk; 978-981-4324-30-4/set; 978-981-4324-35-9/ebook). 1099-1125 (2011).
Let \(X\) be a moduli space for some moduli functor defined in characteristic zero. It is well known that a formal neighbourhood of any point of \(X\) is controlled by some differential graded Lie algebra. There are many examples of this principle (see, e.g., [S. Kosarew, “Local moduli spaces and Kuranishi maps”, Manuscr. Math. 110, No. 2, 237–249 (2003; Zbl 1102.32300)] for some geometric examples).
The main aim of the paper under review is to give a precise meaning to the above principle in terms of higher category theory (see [J. Lurie, Higher topos theory. Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)]). The author traces the above principle to the paper of D. Quillen in [“Rational homotopy theory”, Ann. Math. (2) 90, 205–295 (1969; Zbl 0191.53702)] and some unpublished work of Deligne, Drinfeld and Feigin.
The problem of formalizing the above principle is considered in several cases. First, the author considers classical moduli problems: global (functor from the category of commutative rings) and formal (corresponding to local Artin algebras). However, the main part of the paper is devoted to derived moduli problems: global (\(E_{\infty}\)-ring spectra) and formal (\(E_n\)-rings). The main theorems are stated as equivalences of categories between the category of moduli problems over a field of characteristic zero and \(\infty\)-categories of differential graded Lie algebras (or their analogues).
For the entire collection see [Zbl 1220.00032].

MSC:

55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
14D15 Formal methods and deformations in algebraic geometry
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
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