The completion of d-abelian categories.

Let A be a finite-dimensional algebra, and M be a d -cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d -cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d -abelian category, known to be equivalent to a d -cluster tilting subcategory of an abelian category A. The completion of M , denoted by Ind (M) , is defined as the universal completion of M with respect to filtered colimits. We explore Ind (M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M , comprising left d -exact functors. Notably, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. For a d -cluster tilting subcategory M of mod A , M → consists of all filtered colimits of objects from M , is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M → is a d -rigid subcategory remains unanswered. However, if it is indeed d -rigid, it qualifies as a d -cluster tilting subcategory. In the case d = 2 , employing cotorsion theory, we establish that M → is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M → is a d -cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]