Subsection 1.1. (0MV7) · ScalingStacks

1.1. Preenvelopes and perpendicular categories

We recall the following definition from [8].

Definition 1.1. Let $\mathcal{F}$ be a full subcategory of a category $\mathcal{C}$ and suppose $X$ is an object of $\mathcal{C}$ . A morphism $\phi:X\rightarrow F$ with $F\in\mathcal{F}$ is called an $\mathcal{F}$ -preenvelope if for each morphism $X\rightarrow F^{\prime}$ with $F^{\prime}\in\mathcal{F}$ there exists a morphism $F\rightarrow F^{\prime}$ making the following triangle commute.

An $\mathcal{F}$ -preenvelope is sometimes called a left $\mathcal{F}$ -approximation. We obtain the notion of an $\mathcal{F}$ -precover by dualising the definition above.

0MU1

Definition 1.2. Let $S$ be an object of a triangulated category $\mathcal{T}$ . The subcategory right $n$ -perpendicular to $S$ , denoted by $S^{\perp_{n}}$ , is given by:

S^{\perp_{n}}:=\{X\in\mathcal{T}\ |\ \textnormal{Hom}(S,\Sigma^{i}X)=0% \textnormal{ for }i=1,\ldots,n\}.

The subcategory right $\infty$ -perpendicular to $S$ , denoted by $S^{\perp_{\infty}}$ , is given by:

S^{\perp_{\infty}}:=\{X\in\mathcal{T}\ |\ \textnormal{Hom}(S,\Sigma^{i}X)=0% \textnormal{ for }i>0\}.

Similarly, one can also define the subcategories left $n$ -perpendicular and left $\infty$ -perpendicular to $S$ .

For a subcategory $\mathcal{S}$ of $\mathcal{T}$ , we define:

	$\displaystyle\mathcal{S}^{\perp}$	$\displaystyle:=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}(S,X)=0\textnormal{ for all% }S\in\mathcal{S}\},$
	$\displaystyle{}^{\perp}\mathcal{S}$	$\displaystyle:=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}(X,S)=0\textnormal{ for all% }S\in\mathcal{S}\}.$

David Pauksztello

Original source: arXiv:0705.0102v2