ScalingStacks

The following example is taken from [7]. Let $𝒞$ be an additive category and let $𝒦(𝒞)$ be its homotopy category. We claim that the following pair of full subcategories of $𝒦(𝒞)$ forms a co-$t$-structure on $𝒯=𝒦(𝒞)$. Let

	$𝒜$	$=$
	$ℬ$	$=$	$\{\text{complexes in }𝒦(𝒞)\text{ isomorphic to complexes }C|C^i=0\text{ for }i>0\}.$

It is clear that $𝒜$ and $ℬ$ are closed under direct summands, and that ${\mathrm{\Sigma }}^{-1}𝒜\subseteq 𝒜$ and $ℬ\subseteq {\mathrm{\Sigma }}^{-1}ℬ$. It is also clear that $\text{Hom}({\mathrm{\Sigma }}^{-1}𝒜,ℬ)=0$. We need to show that property $(3)$ of Definition 1.4 holds. Suppose $X$ is an object of $𝒦(𝒞)$:

We obtain the following semi-split short exact sequence of complexes:

which gives us a distinguished triangle

$${\mathrm{\Sigma }}^{-1}A\to X\to B\to A$$

in $𝒦(𝒞)$. Hence $(𝒜,ℬ)$ is a co-$t$-structure on $𝒯=𝒦(𝒞)$. Moreover, it is non-degenerate and its coheart is just the class of complexes sitting in degree zero.