Clarifying a possible error in wikipedia regarding generation of t-structures I am very much a beginner to this subject but I can't make sense of something in the Wikipedia page for t-structures. I am thinking it is an error but I not qualified enough to trust that I'm not just mistaken.
The page is here. Specifically, the claim there is that given a triangulated category $\mathcal{D}$ with arbitrary coproducts, then a set of compact objects generates a t-structure whose co-aisle is given by
$$
\mathcal{D} = \{ X \in \mathcal{D}: \text{Hom} (S_{0}[-n], X) = 0 , n \geq 0  \}.
$$
My issue is that this co-aisle doesn't seem to be closed under negative suspensions, but only positive suspensions. I think it should be $n < 0$. Or otherwise instead of $-n$ replace it with $n$ and change the inequality to be a strict one.
Am I correct in that Wikipedia seems to have an error here?
 A: The definition given of $D^{\geq 1}, D^{\leq 0}$ are correct as written (when using the homological indexing convention). The use of terminology may be slightly confusing though: the t-structure they are describing is $(D^{\leq 0},D^{\geq 0})$, where the co-aisle is $D^{\geq 1}[-1]$ in the homological indexing convention. I.e. the coaisle is:
$$D^{\geq 0} = \{X\in \mathcal{D}\mid \text{Hom}(S_0[-n],X) = 0, n\geq 1\}.$$
That this is a t-structure follows directly, for example, from Lemma 3.1 and Theorem A.1 of [AJS].
The aisle and coaisle are not required to be closed under arbitrary shifting. In fact an aisle is only closed under arbitrary shifting if it is a triangulated subcategory. In the literature, t-structures whose aisles are triangulated subcategories are sometimes referred to as Bousfield localisations.
[AJS] L. Alonso Tarrío, A. Jeremías López, M.J. Souto Salorio, Construction of t-structures and equivalences of derived categories, Trans. Amer. Math. Soc. 355 (2003) 2523–2543.
