# Tamarkin’s paper, chapter 4

### One comment

1. chriskuo says:

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})\$