Reduction for $(n+2)$-rigid subcategories in extriangulated categories

Abstract: In this work we study how to extend the concept of \emph{"reduction,"} given for rigid and functorially finite subcategories in an extriangulated category $\mathcal{C}$, to $(n+2)$-rigid ones. We define the reduction of such subcategories as the intersection of orthogonal complements when certain orthogonal condition is satisfied and we prove that this reduction depends mainly on the subcategory itself beyond the type of extriangulated category for which belongs to. Specifically, we show that some results proven for Frobenius extriangulated categories can be carried to extriangulated categories in general. We also study its properties among we can mention: weakly idempotent completeness, existence of enough $\mathbb{E}$-projectives and $\mathbb{E}$-injectives, and conditions to be Frobenius. That generalization covers the usual case and for the general case (for $n\geq 0$), we provide several examples. Finally, we extend a well-known result given for the reduction in stably $2$-Calabi Yau Frobenius extriangulated categories related with $2$-cluster tilting subcategories.