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1. 

   May 29, 2017 at 11:05 am

   chriskuo says:

   In chapter 4, Tamarkin defines $s{h}_q(X)$ and its quasi-classical reduction $s{h}_{1/2^n}(X)$ for a locally compact topological space $X$. Before defining them, we first need to define some categories which they are enriched over.We define two categories whose objects are $ϵ\cdot \mathbb{Z}$-graded object in $\mathcal{C}$ with morphism complexes
   $$\mathrm{Hom}(X,Y)=\prod _{k\le l}{\mathrm{Hom}}_{\mathcal{C}}({\mathbf{gr}}^{k}X,{\mathbf{gr}}^{l}Y)$$ and
   $$\mathrm{Hom}(X,Y)=\prod _k{\mathrm{Hom}}_{\mathcal{C}}({\mathbf{gr}}^{k}X,{\mathbf{gr}}^{k}Y)$$
   where ${\mathbf{gr}}^{k/2^n}X$ is the $k/2^n$-th graded component of $X$. The categories $\mathbf{Com}(\mathcal{C})⟨1/2^n⟩$ and $\mathbf{Gr}(\mathcal{C})⟨1/2^n⟩$ are constructed from the above two categories by twisting with differentials. ($\mathcal{C}$ to $\mathbf{D}\mathcal{C}$.)

   Let us denote the dg category of chain complex of $\mathbb{Q}$-vector spaces by $\mathfrak{A}$.
   We can define categories $\mathbf{Quant}(\mathcal{C})⟨1/2^n⟩$, $\mathbf{Quant}(\mathcal{C})$ and $\mathbf{Classic}(\mathcal{C})⟨1/2^n⟩$ enriched over $\mathbf{Com}(\mathfrak{A})⟨1/2^n⟩$, $\mathfrak{A}$ and $\mathbf{Gr}(\mathfrak{A})⟨1/2^n⟩$.
   The category $s{h}_q(X)$ is defined as a subcategory of $\mathbf{Quant}((\mathbf{D}⨁){\mathrm{Open}}_X^{\mathbf{op}})$ and $s{h}_{1/2^n}(X)$ is a subcategory of $\mathbf{Classic}((\mathbf{D}⨁){{\mathrm{Open}}^{\mathbf{op}}}_X)⟨1/2^n⟩$.

   For example, without the differentials, the objects of $\mathbf{Quant}(\mathcal{C})$ are $\mathbb{R}$-graded objects in $\mathcal{C}$ and for any $X$, $Y$,
   $$\Hom(X,Y) = \prod_{c \in \mathbb{R} } \prod_{k \geq 0} \bigoplus_{0\leq \delta 0}$isaweakequivalence.When$X = \{ \ast \}$,$\mathbf{Quant}((\mathbf{D} \bigoplus) {\operatorname{Open}^\mathbf{op}}_X) = \mathbf{Quant} (\mathfrak{A} )$sothereshouldbeafamilyofchaincomplexes$T$labeledby$\mathbb{R}$suchthat$\eta(T) \sim \mathbb{Q}_{[a,b)}$.Whatdoesthiskindof$T$ look like?

   2. The functor $\eta$ is defined by a homotopy colimit and they are defined by the bar construction in this paper. Since that’s a large amount of data, is there some way to simplify the chain complexes we get from this construction in this $\eta$ case? Or maybe we can understand it by some universal property instead of the construction?

   3. Since $s{h}_q(X)$ is weak equivalent to $sh(X\times \mathbb{R}{)}_{>0}$ which is something more classic, maybe we have some classic interpretation of the quasi-classical reduction $s{h}_{1/2^n}(X)$ as well?

   Note:
   Tamarkin defines $\mathbf{D}\mathcal{C}$ for a category $\mathcal{C}$ enriched over $\mathfrak{A}$. But he didn’t really define what does that mean for $\mathbf{D}\mathcal{C}$ when $\mathcal{C}$ is enriched over $\mathbf{Com}(\mathfrak{A})$