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Secret Microlocal Seminar

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Tamarkin’s paper, chapter 3

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Let $K$ be a compact set in a locally compact topological space $X$. In this chapter, Tamarkin defines $\mathbb{Z}_K$ by the homotopy limit of \v{C}ech complexes formed by its finite open covers. My question is in the case $K$ can be associated with some good stratification, for example, $[a,b] \subset \mathbb{R}^1$, is this object the same as the \v{C}ech complex associated to the stratification?