Proposition 2.6 (0MXD) · ScalingStacks

0MXD

Proposition 2.6. Let $\mathcal{C}$ be an idempotent complete triangulated category with bounded weight structure $w$ . Assume that $\mathcal{C}=\operatorname{h}\!\mathcal{C}_{\infty}$ arises as the homotopy category of a stable $\infty$ -category $\mathcal{C}_{\infty}$ . Then there is a functor of triangulated categories called weight complex functor

(2.1)

\displaystyle t:\mathcal{C}\to\operatorname{K}^{b}(\mathcal{C}^{w=0})

that restricts to the natural embedding $\mathcal{C}^{w=0}\to\operatorname{K}^{b}(\mathcal{C}^{w=0})$ into degree $0.$

0MXE

Proof. We apply Proposition 2.5 to $\mathcal{C}_{\infty}$ and $\mathcal{D}_{\infty}=\operatorname{N}\!\operatorname{Ch}\!^{b}(\mathcal{C}^{w=0}),$ the stable $\infty$ -category of bounded chain complexes with values in $\mathcal{C}^{w=0}$ from [Lur09, Example 4.4.5.1] and [Lur12, Section 1.3.1]. The homotopy category $\operatorname{h}\!\mathcal{D}_{\infty}=\operatorname{K}^{b}(\mathcal{C}^{w=0})$ of $\mathcal{D}_{\infty}$ is the bounded homotopy category of chain complexes in $\mathcal{C}^{w=0}$ and equipped with the canonical weight structure with heart $\mathcal{C}^{w=0},$ see Example 2.2(1). The heart of the $\infty$ -category $\mathcal{D}_{\infty}$ is $\mathcal{D}_{\infty}^{w=0}=\operatorname{N}\!(\mathcal{C}^{w=0}),$ the nerve of $\mathcal{C}^{w=0}.$

The unit of the adjunction between the nerve and homotopy category functors yields a functor $\epsilon:\mathcal{C}_{\infty}^{w=0}\to\mathcal{D}_{\infty}^{w=0}=\operatorname{N}\!(\operatorname{h}\!\mathcal{C}_{\infty}^{w=0}).$ The $\infty$ -categorical weight complex functor $t_{\infty}:\mathcal{C}_{\infty}\to\mathcal{D}_{\infty}=\operatorname{N}\!\operatorname{Ch}\!^{b}(\mathcal{C}^{w=0})$ is the unique (up to equivalence) weight exact functor such that $\operatorname{Res}(t_{\infty})=\epsilon,$ where $\operatorname{Res}$ is the functor defined in Proposition 2.5. On the homotopy categories, $t_{\infty}$ induces the weight complex functor $t$ . ∎