Geodesic
The hearts of weight structures are the weakly idempotent complete categories
Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 29-38

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This paper proves that additive categories that occur as hearts of weight structures are precisely the weakly idempotent complete categories, that is, the categories where all split monomorphisms give direct sum decompositions. The work also gives several other conditions equivalent to weak idempotent completeness (some of them are completely new) and discusses weak idempotent completions of additive categories.
Keywords: weakly idempotent complete category, idempotent completion, weak retraction-closure, triangulated category, weight structure, heart. 
@article{CHEB_2020_21_3_a5,
     author = {M. V. Bondarko and S. V. Vostokov},
     title = {The hearts of weight structures are the weakly idempotent complete categories},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {29--38},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2020_21_3_a5/}
}
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M. V. Bondarko; S. V. Vostokov. The hearts of weight structures are the weakly idempotent complete categories. Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 29-38. http://geodesic.mathdoc.fr/item/CHEB_2020_21_3_a5/