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Higher homological algebra was introduced by Iyama. It is also known as   $n$  -homological algebra where   $n \geqslant 2$   is a fixed integer, and it deals with   $n$  -cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with   $n+2$   objects. This was recently formalized by Jasso in the theory of   $n$  -abelian categories. There is also a derived version of   $n$  -homological algebra, formalized by Geiss, Keller, and Oppermann in the theory of   $(n+2)$  -angulated categories (the reason for the shift from   $n$   to   $n+2$   is that angulated categories have  tri angulated categories as the “base case”). We introduce torsion classes and t-structures into the theory of   $n$  -abelian and   $(n+2)$  -angulated categories, and prove several results to motivate the definitions. Most of the results concern the   $n$  -abelian and   $(n+2)$  -angulated categories   $\mathscr {M}(\Lambda )$   and   $\mathscr {C}(\Lambda )$   associated to an   $n$  -representation finite algebra   $\Lambda $  , as defined by Iyama and Oppermann. We characterize torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in   $\mathscr {M}(\Lambda )$   and intermediate t-structures in   $\mathscr {C}(\Lambda )$   which is a category one can reasonably view as the   $n$  -derived category of   $\mathscr {M}(\Lambda )$  . We hint at the link to   $n$  -homological tilting theory.

Torsion Classes and t-Structures in Higher Homological Algebra