The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan formalism indeed produces a deformation problem and conversely for most deformation problems the dg-Lie algebra is well known. In the derived setting this principle has even become a theorem following Lurie-Pridham. However, I was wondering if anyone could explain the original intuition Deligne had behind this principle?
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5$\begingroup$ In case it could be helpful, here is the link to the pdf file with the scanned copy of the referred letter of Deligne to Millson: publications.ias.edu/sites/default/files/millson.pdf $\endgroup$– agtCommented Sep 15, 2016 at 19:28
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3$\begingroup$ From a modern perspective the point is that deformation problems are "formal pointed stacks," and it should be possible to understand these by understanding their "loop spaces." In other words, the sense in which a dg Lie algebra controls a deformation problem is that the deformation problem is the "formal classifying stack" of the dg Lie algebra. This is a version of Koszul duality, the same version that powers the appearance of dg Lie algebras in rational homotopy theory. I don't know if this was Deligne's intuition. $\endgroup$– Qiaochu YuanCommented Sep 15, 2016 at 21:06
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2$\begingroup$ If you want to get the intuition, take a look at the first chapter of books.google.com.br/… . You may look at the chapter on deformation theory in Huybrechts book on complex geometry. The Maurer Cartan equation classifies the obstruction to lift a deformation of some order to a higher... $\endgroup$– user40276Commented Sep 16, 2016 at 1:55
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$\begingroup$ order. In other words, the full Maurer-Cartan equation gives you a formal deformation. In modern general terms, this is Koszul duality: $T\Omega X = T[1]X $ is an L infinity algebra for an infinity stack $X$, by viewing $\Omega X$ as an infinity group. $\endgroup$– user40276Commented Sep 16, 2016 at 1:57
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I suspect, Deligne's intuition went along the following lines. Deformation theory describes the tangent cone at a point $x$ of the moduli space $M$ of the problem. The tangent cone is Spec Gr $\mathcal{O}_{M,x}$, where the associated graded Gr is taken with respect to powers of the maximal ideal of the local ring $\mathcal{O}_{M,x}$, the stalk of the structure sheaf $\mathcal{O}_{M}$ at $x$. Usually, this cone is singular. You may resolve this singularity within derived algebraic geometry, for instance, find a free dg-commutative algebra whose cohomology is Gr $\mathcal{O}_{M,x}$. A free dg-commutative algebra is equivalent to an $L_\infty$-algebra. Then you take a quasi-isomorphic dg-Lie algebra, and you are done.
