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have created a “floating TOC” notions of subcategory and have included it in the relevant entries
Maybe a better and simpler title would be simply “subcategories” (this is a topic and it is potentially a bit wider than listing kinds of subcategories, for example it would include torsion theory which is a couple of subcategories satisfying some axioms). I think we also have entry thick subcategory.
But here is somewhat related genuine question, not for subcategories but subfunctors. There is something I called in one paper “equivalence of subfunctors” and I did not know the standard term. Maybe somebody does ? So let F1,F2:A→B be subfunctors of a functor G:A→B in which B is a category with images. We say that F1 is equivalent to F2 if there is a natural automorphism α:G⇒G such that for every a∈Ob(A) the restriction αa|ImF1(a) is isomorphism onto ImF2(a). Thus, roughly speaking there is an natural isomorphism of functors F1≅F2 which extends to a natural automorphism of G.
I have added thick subcategory and Serre subcategory and topologizing subcategory to the list.
added bireflective subcategory and touched the formatting of the first few lines
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