Mac Lane defines a subcategory as a subset of objects and a subset of morphisms that form a category. But the first rule of category theory is that you do not talk about equality of objects. Up to equivalence, the definition becomes a faithful functor. This is a useful concept, but I don't think it fits the name. I don't want groups to be a subcategory of sets!


This is not a question about aesthetics, but about usage. I don't think people tend to use Mac Lane's definition. Maybe they're just wrong, but I'd like to know if there is another definition which fits the usage better. [All the time](https://mathoverflow.net/questions/23515/cofinal-inclusions-of-waldhausen-categories) I see people say things like "we may assume that our subcategory contains every object isomorphic to an object of the subcategory." I guess we can expand to an equivalent subcategory to achieve this, but we probably have to choose *how* the objects are isomorphic (though it may be easier if to change the ambient category). This is a much more natural thing to do (and safer) if the subcategory is full, or at least contains all the automorphisms of its objects. This leads me to suspect that people are assuming or thinking of some stronger definition than faithful.


Do people tend to mean the official definition? or do they also require full? containing all the automorphisms? Are there other useful intermediate notions?



What do people mean by "subcategory"?
Do you want the notion of "subcategory" to be invariant under categorical equivalence? If so, then "pseudomonic" functors are the right thing: faithful, and full on isomorphisms.


But I don't think one would want this any more than the notion of "inclusion" of topological spaces to be invariant under homotopy equivalence (which would make it meaningless).


