Dear all, Let me first thank the persons who have answered my questions on terminology. Since the answers were different, it seems that the terminology is not standard and that my questions made sense. Preliminary remarks. By now, everybody understands what kind of categories I was talking about. they are very simple, one might be tempted to say trivial. But in many highly non trivial questions they appear either as "building bricks" of more complex constructions (see e.g. fibrations such that all the fibers are of that kind), or as special cases, unavoidable, of more general situations (e.g. Freyd's notion of "equivalence kernels"). Of course such categories can be "internalized ", say in a topos, (this is much too strong), and it would be nice that the terminology should fit also the internal case, and in particular any reference, explicit or implicit, to AC should be avoided. 1- Discrete versus indiscrete, or coarse, etc. The categories 0 and 1 are both discrete and indiscrete So each name night pose a problem. The terminology "discrete" is by now well established. Thus I think one should avoid "indiscrete" and use "coarse" instead. Thus 0 and 1 are discrete and coarse and that is admissible. In the internal case every sub object of 1 is both discrete and indiscrete. So "subterminal" is a nice name to cover both cases. 2- "Essentially". One has to be careful about this word. It seems to mean "up to equivalence". But that depends on what you call "equivalence of categories". There is a very strong 2-categorical notion, namely a pair of functors f: A --> B and g: B --> A with fg and gf isomorphic to identities (with or without the adjunction axioms). But it is useless for our purpose. The one which might serve here is f full and faithful and essentially surjective. But unless we have AC it is not symmetric, even for A and B small. Thus with AC we might adopt the suggestions of Thomas and use essentially discrete and essential sub terminator. But do we really need AC or even any notion of equivalence at all? 3- Elementary remarks. Let S be a category with finite limits. I want to "internalize" the two notions for which I'd like a name, suitable not only when S=Set, but in general. (i) if X is an internal category I denote by Ob(X) and Map(X) the objects of objects and maps of X and by d: Map(X) --> Ob(X)xOb(X) with projections Dom and Codom. When S=Set if f: x -->y is a map of X, df is the "direction" of f. X is a preordered object iff d is a mono. It is easy, using the multiplication of X, to express in terms of finite limits, the fact that X is a groupoid. Hence equivalence relations are definable in any category with finite limits, although composition of arbitrary relations is possible only when S is regular. Moreover if F: S --> S' is a functor which preserves finite limits and X is an equivalence relation so is F(X) No essentiality no AC is needed (ii) the functor X --> 1 is full and faithful iff the direction map d is an iso. This again is preserved by finite limits preserving functors F. (iii) If F preserves finite limits and is faithful it reflects equivalence relations and property (ii). In particular the Yoneda embedding preserves and reflects the previous properties. this enables us to work with them as if S=Set. No AC is involved . For all these reasons I'm not too keen about using "essentially" I have many more remarks about Fred Linton's and David Robert's postings but this mail is already a bit long. So I shall wait a few days before i send another mail. By that time I hope to have some reactions to the present posting and I shall do my best to answer them. Best regards, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]