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

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We are having a small discussion group around Tamarkin’s paper “microlocal category”.  There will be a blog post for each chapter, in which we collect together our understandings and misunderstandings.  At the moment they are mostly empty. This is the post for chapter 1.  

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Question: What’s the sheaf theoretic incarnation of this filtration? (Warmup question: what structure does a filtration on a DGA induce on its representation category?)

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Let $M$ be a manifold; $shv(M)$ the (derived) category of sheaves on $M$, and $\mathcal{G}(M)$ the group of autoequivalences of $shv(M)$. I want to consider the following sort of right-invariant semi-metric (i.e. some things have distance zero) on $\mathcal{G}(M)$: $$d(\gamma ,\eta )=su{p}_F{d}_{Haus}(ss(\gamma F),ss(\eta F))$$ Here, we take $ss(F)$ in the cosphere bundle …

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