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.



structure in triangulated category similar to t-structure
In


*Jørgensen, Peter*, [**Abelian subcategories of triangulated categories induced by simple minded systems**](https://doi.org/10.1007/s00209-021-02913-5), Math. Z. 301, No. 1, 565-592 (2022). [ZBL1503.16015](https://zbmath.org/?q=an:1503.16015).


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](https://www3.nd.edu/%7Edyer/papers/extri.pdf)).


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.


