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Tamarkin’s paper, chapter 4

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In chapter 4, Tamarkin defines $sh_q(X)$ and its quasi-classical reduction $sh_{1/2^n} (X)$ 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 $\epsilon \cdot \mathbb{Z}$-graded object in $\mathcal{C}$ with morphism complexes

$$ \operatorname{Hom}(X, Y) = \prod_{k\leq l} \operatorname{Hom}_{ \mathcal{C} } (\mathbf{gr}^k X,\mathbf{gr}^l Y) $$ and

$$ \operatorname{Hom} (X, Y) = \prod_{ k} \operatorname{Hom}_{ \mathcal{C} } (\mathbf{gr}^k X,\mathbf{gr}^k Y) $$

where $\mathbf{gr}^{k / 2^n} X $ is the $k / 2^n$-th graded component of $X$. The categories $\mathbf{Com}(\mathcal{C}) \langle 1/2^n \rangle$ and $\mathbf{Gr} (\mathcal{C})\langle 1/2^n \rangle$ 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 $\mathbf{Quant}(\mathcal{C})\langle 1/2^n \rangle$, $\mathbf{Quant}(\mathcal{C})$ and $\mathbf{Classic}(\mathcal{C})\langle 1/2^n \rangle$ enriched over $\mathbf{Com}(\mathfrak{A}) \langle 1/2^n \rangle$, $\mathfrak{A}$ and $\mathbf{Gr} (\mathfrak{A}) \langle 1/2^n \rangle$.

The category $sh_q(X)$ is defined as a subcategory of $\mathbf{Quant}((\mathbf{D} \bigoplus){\operatorname{Open}}^\mathbf{op}_X)$ and $sh_{1/2^n}(X)$ is a subcategory of $\mathbf{Classic}((\mathbf{D} \bigoplus) {\operatorname{Open}^\mathbf{op} }_X)\langle 1/2^n \rangle$.

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}$ is a weak equivalence. When $X = \{ \ast \}$, $\mathbf{Quant}((\mathbf{D} \bigoplus) {\operatorname{Open}^\mathbf{op}}_X) = \mathbf{Quant} (\mathfrak{A} )$ so there should be a family of chain complexes $T$ labeled by $\mathbb{R}$ such that $\eta(T) \sim \mathbb{Q}_{[a,b)}$. What does this kind of $T$ 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 $ sh_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 $sh_{1/2^n} (X)$ 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 $\mathbf{Com}(\mathfrak{A})$