What’s the point of always adjoining Maurer-Cartan elements?

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Secret Microlocal Seminar

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Tamarkin’s paper, chapter 2

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What’s the point of always adjoining Maurer-Cartan elements?

Does anyone know any $F$ and $I$ that can result some familiar things for $\operatorname{hocolim}_I F$?

I guess one of the reason why he adjoins Maurer-Cartan elements (at lease for $Quant \langle \epsilon \rangle $) is in the discussion in section1.1. What he said there is that we want to give $D_{>0} (X\times \mathbb{R})$ a dg-structure for some quantization reason. More precisely, for $F$, $G \in D_{>0}(X \times \mathbb{R})$, we want to give $\underline{Hom}(F,G)$ a $\mathbb{Q}[q]$-module structure. But the category of $\mathbb{Q}[q]$-modules has problems and we need a full sub-category of ‘semi-free’ objects to work with. (A ‘semi-free’ chain complex is a chain complex having free object in each degree.) And he claim $Quant \langle \epsilon \rangle$ which consists of object having the form $(X,D_X)$ where $X$ is a object already in a dg-category consisting of $\mathbb{Q}$ vector spaces and $D_X$ is a Maurer-Cartan element is the category we’re looking for. (See p.11)