On the maximal exact structure of an additive category
We prove that every additive category has a unique maximal exact structure in the sense of Quillen.
We prove that every additive category has a unique maximal exact structure in the sense of Quillen.
It was recently proved that every additive category has a unique maximal exact structure, while it remained open whether the distinguished short exact sequences of this canonical exact structure coincide with the stable short exact sequences. The question is answered by a counterexample which shows that none of the steps to construct the maximal exact structure can be dropped.
Let be a semibrick in an extriangulated category. If is a -semibrick, then the Auslander-Reiten quiver of the filtration subcategory generated by is . If is a -cycle semibrick, then is .