Deformation theory of objects in homotopy and derived categories. II: Pro-representability of the deformation functor
In the first part [\textit{A. I. Efimov, V. A. Lunts} and \textit{D. O. Orlov}, Adv. Math. 222, No.~2, 359--401 (2009; Zbl 1180.18006)] of this series of articles the authors developed a general deformation theory of objects in both the homotopy and derived category of differential graded categories. This second part is concerned with representability issues of the (derived) pseudo-functors Def\((E)\) and coDef\((E)\) defined on the category of artinian DG algebras for a DG module \(E\) over a DG category. Since they take values in the \(2\)-category of groupoids, rather than sets, such a question is better dealt with if the source is a \(2\)-category of artinian DG algebras. There are actually two such natural bicategories, one on which an extension of the codeformation can be defined, the other where the deformation can be extended. These two \(2\)-categories are equivalent and the deformation DEF\((E)\) and codeformation coDEF\((E)\), when restriced to negative artinian DG categories, coincide under some extra finiteness conditions on \(E\). The main pro-representability theorem states then that these equivalent pseudo-functors are pro-represented by the dual of the bar construction on the minimal \(A_\infty\)-model of \({\mathbb R} \text{Hom}(E, E)\). This motivates the study in the first part of this paper of \(A_\infty\)-algebras, categories, and modules, as well as bar constructions and the Maurer-Cartan pseudo-functor in this setting.
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