symmetric monoidal (∞,1)-category of spectra
Deformation theory studies problems of extending structures to extensions of their domains. Formal deformation theory, is the part of the deformation theory where the extensions are infinitesimal.
A typical problem in formal deformation theory has the structure that
and infinitesimal thickenings $\tilde X$ and $\tilde Y$ of $X$ and $Y$ are prescribed, with injection morphisms $X \to \tilde X$ and $Y \to \tilde Y$
and asks whether a bottom horizontal morphism $\tilde f$ in the diagram
may be found. This morphism $\tilde f$ would be called an infinitesimal deformation of $f$.
In other words:
Formal deformation theory studies the obstruction theory of extensions to infinitesimal thickenings.
A typical example of an infinitesimal thickening is a square-0-extension of a ring:
let $R$ be a ring, to be thought of as the ring of functions on the space $X$ in the above diagram. Let furthermore $N$ be an $R$-module, to be thought of as the $R$-module of sections of a vector bundle over $X$.
Then consider the new ring, whose underlying group is the direct sum $R \oplus N$, equipped with the product structure
This is the square 0-extension of $R$ by $N$. It should be thought of as the algebra of functions that consists of elements of $R$ and $N$, where the elements in $N$ are thought of as functions with values in infinitesimal quantities, so that their would-be product “$n_1 \cdot n_2$” vanishes.
So the ring $R \oplus N$ may be thought of as the ring of functions on the infinitesimal extension $\tilde X$ of $X$, which is the space obtained by adding to $X$ all the vectors of infinitesimal length in the vector bundle over $X$.
There is a canonical ring homomorphism $R\oplus N \to R$ that is the identity on $R$ and $0$ on $N$. This is to be thought of as the pullback of functions on spaces along the inclusion of spaces $X \to \tilde X$ (which in turn may be thought of as the 0-section of the vector bundle on $X$).
Similarly, let $R_2$ be another ring with module $N_2$ and square-0 extension $R_2 \oplus N_2$, thought of, respectively, as the ring of functions on a space $Y$, the module of sections of a vector bundle on $Y$ and the ring of functions on the space of infinitesimal vectors of this vector bundle.
In terms of these function rings, a morphism $f : X \to Y$ of spaces corresponds to a ring homomorphism $R_1 \leftarrow R_2 : f^*$. Hence we have a situation
The obvious obstruction problem now is whether we can deform $f^*$ to a morphism $R_1 \oplus N_1 \leftarrow R_2 \oplus N_2 : \tilde f^*$ of rings, such that we get a commuting diagram
The obstruction to the existence of such lifts is measured by cohomology with coefficients in the cotangent complex of $R_1$.
This is the archetypical problem that deformation theory deals with. As always, after studying this a bit it turns out that in order to obtain a good theory, one needs to adopt the nPOV. Problems as above may be stated in the category Ring of rings, but they may have good answers only in categorifications of this for instance to the (∞,1)-category of E-∞-rings.
In order to better see the structure of the above archetypical problem of deformation theory, we describe some aspects of the canonical bifibration of ring modules in a way that nicely organizes all the concepts module, derivation, Kähler differential in a single picture that lends itself to vertical categorification. (Following DefTheory.)
With Ring denoting the category of (commutative, unital) rings, write
for the bifibration of modules over rings: objects of $Mod$ are pairs consisting of a ring $R$ an an $R$-module $N$, and morphism $(R_1,N_1) \to (R_2, N_2)$ are pairs consisting of a ring homomorphism $f : R_1 \to R_2$ and a morphism $F : N_1 \to N_2 \otimes_f R_2$ of $R_2$-modules.
(Recall for instance from the discussion at Sweedler coring) that this bifibration is a way to think of the stack of algebraic vector bundles.)
But there is also another functor $G : Mod \to Ring$ of interest: for $N$ any $R$-module, we may form the ring $G(N) := R \oplus N$ called the square 0-extension of $R$, in which multiplication is given by
Moreover, there is a natural morphism of rings $G(N) \to R$ given by sending $(r,n) \mapsto r$. A section $v : R \to G(n)$ of this morphism is precisely a derivation of $R$ with values in the module $N$.
This may be organized into a single functor
into the arrow category of Ring, that sends to the $R$-module $N$ to the morphism $G(N) \to R$. The original bifibration factors through this morphism by the right endpoint evaluation
Finally notice that the functor $G$ has a left adjoint functor
that sends a ring $R$ to the $R$-module $\Omega_K(R)$ of Kähler differentials, i.e. to the module that encodes the cotangent bundle.
Using the module of Kähler differentials is not appropriate in general; instead we need to take its derived version. To talk about the nonabelian derived functors, Quillen introduced a model category structure on the category of simplicial commutative rings. Given a morphism $f: A\to B$ of rings, which makes $B$ an $A$-algebra, the category $AbGr(A-Alg/B)$ of abelian group objects in the slice category $A$-$Alg/B$ of $A$-algebras over $B$ is equivalent both to the category of $B$-modules and the trivial (= square zero) extensions of $A$ by $B$-modules. In particular we can consider the forgetful functor $AbGr(A-Alg/B)\to A-Alg/B$ which has a left adjoint $Ab_{B/A} : A-Alg/B\to AbGr(A-Alg/B)\cong {}_B Mod$. All said is true for simplicial commutative rings as well. Now the relative cotangent complex $L_{B/A}$ is the value on $B$ of the left derived functor $\mathbb{L} Ab_{B/A}(B)$. Regarding that the left adjoint at the nonderived level (and for usual rings) can be expressed via Kähler differentials, this explains the phrase “derived version of the module Kähler differentials”.
The above situation generalizes from the category Ring to an arbitrary presentable (∞,1)-category $C$ by replacing the bifibration $Mod \to Ring$ by the stabilization $T_C \to C$ of the codomain fibration of $C$: the tangent (∞,1)-category of $C$.
The projection $p : T_C \to C$ still has a left adjoint
for which a representative which is also a section (in a strict sense) of $p$ may be taken; any such representative is called the cotangent complex functor for $C$. The special property section property, like in the motivating example above, says that the composition
is the identity (∞,1)-functor.
…
Over a field of characteristic zero, there is an approach to deformation theory via differential graded Lie algebras (or more generally L-infinity algebras). One can find some exposition about this approach in the Kontsevich and Lurie references below. See also discussion at MathOverflow: def theory and dgla-s.
In this approach, one begins with an object $X$ (for example a scheme, or a complex manifold, or a vector bundle, or an associative algebra, or a dg category, or …) that one would like to deform. Then the general principle is that there exists a dgLa $L_X$ with the property that the functor $Def_{L_X} : Art \to Set$, which sends a local Artin algebra $(A,m)$ to the set of Maurer-Cartan solutions in $(L_X \otimes m)^1$ modulo the gauge action of $(L_X \otimes m)^0$, is isomorphic to the functor which sends a local Artin algebra $(A,m)$ to the set of isomorphism classes of deformations of $X$ over $\operatorname{Spec} A$. Note the similarity with Schlessinger’s theory: both here and in Schlessinger’s work, we deal with functors from Artin algebras to sets.
In the case of a compact complex manifold, the dgLa in question is given by the so-called Kodaira-Spencer? dgLa: holomorphic vector fields tensor $(0,q)$-forms (this is just the Dolbeault resolution of the sheaf of holomorphic vector fields). In the case of an associative algebra (or a dg algebra, or an A-infinity algebra, or a dg category, or an A-infinity category), the appropriate dgLa is the Hochschild complex with the Hochschild differential and the Gerstenhaber bracket.
In this language, the Tian-Todorov theorem? on the unobstructedness of deformations of Calabi-Yau manifolds translates to the statement that the Kodaira-Spencer dgLa of a Calabi-Yau manifold is homotopy abelian — that is, it is quasi-isomorphic to an abelian dg Lie algebra. Barannikov-Kontsevich proved more generally that the dgLa given by holomorphic polyvector fields tensor $(0,q)$-forms on a Calabi-Yau manifold is homotopy abelian. The deformation-theoretic consequence is that the “extended deformations” of Calabi-Yau manifolds are unobstructed. These “extended deformations” should be realized by certain $A_\infty$ deformations of (a dg enhancement of) the derived category of coherent sheaves on the Calabi-Yau.
The following paper is a good introduction to these ideas:
The Kontsevich and Soibelman references below are also good.
deformation context, formal moduli problem, Lie differentiation
tangent complex, André-Quillen cohomology, Hochschild cohomology
cotangent complex, André-Quillen homology, Hochschild homology
Michael Schlessinger, Jim Stasheff, The Lie algebra structure of tangent cohomology and deformation theory, Journal of pure and applied algebra 38 (1985) 313-322
Michael Schlessinger, Jim Stasheff, Deformation theory and rational homotopy type (arXiv:1211.1647)
C. Doran, Deformation Theory: An Historical Annotated Bibliography (pdf)
M. Doubek, M. Markl, P. Zima, Deformation theory (lecture notes), Archivum mathematicum 43 (5), 2007, 333–371, arXiv:0705.3719
Martin Markl, Deformation theory of algebras and their diagrams, 129 pp, CBMS 116, AMS 2012, book page
Wikipedia: deformation theory, cotangent complex
E. Sernesi, An overview of classical deformation theory, pdf
Alexander Grothendieck, Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes in Mathematics 79
Luc Illusie, Complexe cotangent et déformations I, Lec. Notes Math. 239, Springer 1971, xv+355 pp.; II, LNM 283, Springer 1972. vii+304 xv+355 pp.
Maxim Kontsevich, Yan Soibelman, Deformation theory I (ps); Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I, math.AG/0606241 – two parts of large unfinished books on the subject; Yan Soibelman, Lectures on deformation theory and mirror symmetry (ps)
Maxim Kontsevich, Topics in deformation theory (A rough write up of a Berkeley course, early 90-s), ps
Jacob Lurie, Deformation Theory (arXiv:0709.3091) – describes a very setup for deformation theory over any (∞,1)-category is described. Then as an application the deformation theory of E-∞-rings is developed. An application: J. Lurie, Moduli problems for ring spectra, moduli.pdf.
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homepage of Ravi Vakil’s graduate Stanford class on deformation theory and moduli spaces
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Gregory Ginot, Sinan Yalin, Deformation theory of bialgebras, higher Hochschild cohomology and formality, arxiv/1606.01504