It’s well known that the heart of a t-structure is an abelian category. My question is that can we find some structure on a triangulated category which can “produce” an exact category in analogy with the t-structure? I would be appreciated if someone can answer this question or give me some related references.
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1$\begingroup$ Not at all answering your question, but are you familiar with the construction of the derived category of a quasi-abelian category, due to Schneiders (DOI:10.24033/msmf.389)? At the very least, you probably want to have some way to find the original category back in this case, as quasi-abelian categories are particular instances of exact categories. $\endgroup$– R. van Dobben de BruynCommented Aug 6, 2023 at 12:09
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$\begingroup$ Thank you for your comment. Although actually I want to construct a t-structure-like thing on triangulated categories which may not from derived categories, what you mentioned is still very interesting! $\endgroup$– Yifei ChengCommented Aug 6, 2023 at 12:48
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1 Answer
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In
Jørgensen, Peter, Abelian subcategories of triangulated categories induced by simple minded systems, Math. Z. 301, No. 1, 565-592 (2022). ZBL1503.16015.
The following theorem due to Matthew Dyer is provided:
Let C be a triangulated category and D be an extension closed subcategory of D such that $Hom^{-1}_{D}(M, N)=0$ for all M, N in D. Then C has a natural structure of exact category (with short exact sequences obtained by suppressing the arrows of degree 1 in the distinguished triangles of C with vertices in D). Moreover, there are natural isomorphisms $Ext^{i}_{C}(M, N)\cong Hom^{i}_{D}(M, N)$ for $0 ≤ i ≤ 1$.
(Taken for convenience from Dyer's original paper here).
This is in direct analogy with the heart of a t-structure being an extension closed subcategory such that $Hom_{D}^{-i}(M,N)=0$ for all $i>0$ and $M,N$ in $C$.
I think from here you could construct a structure analogous to a pair of aisles $(D^{\leq 0},D^{\geq 0})$ of a t-structure which gives you these sort of subcategories as 'hearts' in some sense. Of course these have to have weaker axioms, like you can't ask for the aisles to be closed under shifts but I'm not sure this will suffice or if something else breaks along the way that you need to avoid from the axioms.
I hope this is of help.
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$\begingroup$ Thank you for your answer. I guess this is exactly what I want. $\endgroup$ Commented Aug 6, 2023 at 12:49