Proposition 1.6 (0MU5) · ScalingStacks

0MU5

Proposition 1.6. Let $\mathcal{T}$ be a triangulated category and suppose $(\mathcal{A},\mathcal{B})$ is a co- $t$ -structure on $\mathcal{T}$ . We have:

(i) For all objects $X$ of $\mathcal{T}$ there exists an $\Sigma^{-1}\mathcal{A}$ -precover $\alpha:\Sigma^{-1}A\rightarrow X$ .

(ii) For all objects $X$ of $\mathcal{T}$ there exists a $\mathcal{B}$ -preenvelope $\beta:X\rightarrow B$ .

(iii) We have $\Sigma^{-1}\mathcal{A}={}^{\perp}\mathcal{B}$ and $\mathcal{B}=(\Sigma^{-1}\mathcal{A})^{\perp}$ .

(iv) $\mathcal{A}$ is closed under extensions.

(v) $\mathcal{B}$ is closed under extensions.

0MU6

Proof: Properties (i) and (ii) are immediate consequences of the definition of a co- $t$ -structure: the $\Sigma^{-1}\mathcal{A}$ -precover $\alpha:\Sigma^{-1}A\rightarrow X$ and the $\mathcal{B}$ -preenvelope $\beta:X\rightarrow B$ are just the first and second morphisms in the distinguished triangle given by property $(3)$ of Definition 1.4. Property (iii) is a consequence of condition $(0)$ and the orthogonality condition $(2)$ of Definition 1.4, and properties (iv) and (v) are easy consequences of property (iii), see [12]. $\Box$