2016/08/01 by Zhou, Panyue, Zhu, Bin · 4 citations
#18E10 #18E30 #18E40 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1608.00297
Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this article. Let \cal A be an extension closed subcategory of an extriangulated category \cal C. Then the quotient category \cal M:=\calA/\calX carries naturally a triangulated structure whenever (\cal A,\cal A) forms an \cal X-mutation pair. This result unifies many previous constructions of triangulated quotient categories, and using it gives a classification of thick triangulated subcategories of pretriangulated category \calC/\calX, where \cal X is functorially finite in \cal C. When \cal C has Auslander-Reiten translation τ, we prove that for a functorially finite subcategory \cal X of \cal C containing projectives and injectives, \calC/\calX is a triangulated category if and only if (\cal C,\cal C) is \cal X-mutation if and only if τ\underline\cal X=\cal X. This generalizes a result by Jørgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory \cal X of the extriangulated category \cal C, \cal C admits a new extriangulated structure such that \cal C is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present examples that extriangulated categories are neither exact categories nor triangulated categories.