We investigate how to characterize subcategories of abelian categories in terms intrinsic axioms. In particular, we find axioms which generating cogenerating functorially finite subcategories, precluster tilting and cluster categories. As a consequence prove that any d-abelian category is equivalent d-cluster subcategory an category, without assumption on the being projectively generated.

Here the solid arrows represent maps that are given, and the problem is to find a map h commuting in the diagram. Usually the map i is an inclusion, and p is some kind of “bundle map” such as a local product. Note the special cases: (i) if i is an inclusion, E = A, and f = 1A, then the extension problem asks whether A is a retract of X; and (ii) if p is surjective, X=B, and g = 1B, the lifting ...