Re: Terminology

["Jean Bénabou" <jean.benabou@wanadoo.fr>]

Dear Thomas, Dear all,

My definition of poset is: "preordered set". I don't know if there is a general agreement, since some answers seemed to suppose that I meant "partially ordered set". It is because I feared this confusion that i specified by adding: equivalent to a discrete category. 
Of course I knew that they were "equivalence relations", and had also many other simple characterizations. One which I like and need is: X is equivalent to a discrete category iff the functor X --> 1 is faithful  and conservative (i.e. reflects isos) because it has the following generalization:

Let P: X --> S be a prefibration. The following are equivalent:
(i) P is equivalent to a discrete fibration.
(ii) P is faithful and consevative.
(iii) each fiber of P is equivalent to a discrete category.

Thus my question  was not:  what are such categories, for which I knew perfectly many answers, but : is there a well established name for them? 
Suggestions such as "setoids" or "essentially discrete" show that this is not the case.

I don't like very much "setoids", and I am very tempted by "essentially discrete" as Thomas suggested. 
But I shall make my question a bit more difficult. 
What would you call a category X such that the functor X --> 1 is full and faithful? Please don't tell me what they are, I  know that. I'm not even asking if there is a we'll established name for them. I don't think there is one. What I ask is: Could you suggest one? Preferably a name which would be suitable when we work with categories internal to a Topos E where supports don't split.

As a side remark, let me say that I don't care very much for the distinction between "evil" and "non evil". Apart  from obvious moral or philosophical reasons, for the following purely mathematical one: Non-evilness depends on the notion of equivalence of categories. And this in turn may heavily depend on which notion of equivalence you chose. And some of these notions depend on the axiom of choice, which I might be tempted to call "evil". Thus we'd reach the following conclusion: 
Non evil is essentially evil. 
I rather like this conclusion, don't you?

Best regards,
Jean

Le 27 avr. 2013 à 15:08, Thomas Streicher a écrit :

> Dear Jean,
> 
a
> 
>> As many of you I presume, I have for ages, and very often, had to deal with categories which are both groupoïds and posets, or again which are equivalent to a discrete category. Is there a well established name for them?
> 
> What about "essentially discrete" like in "essentially small" or
> "essentially surjective". Generally, for any property P of categories
> I would say a category is "essentially P" if it is equivalent to a
> category with property P.
> So "essentially" is a kind of magic word transforming "evil" properties
> into "non-evil" ones. (I don't think one should always do this!)
> 
> Thomas



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