Re: Terminology

[Toby Bartels <categories@TobyBartels.name>]

Thomas Streicher wrote:

>Jean B?nabou wrote:

>>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.

>Sticking to the pattern I suggested I'd call it "essentially subterminal".

I learnt to call that an "indiscrete category", so I probably would.
(Another term that I've heard is "chaotic category", which I never liked.)
Of course, I could also call it a "truth value",
but only in a context where I would expect this to be understood
(and being "non-evil", that is working up to equivalence,
is not actually sufficient for that).  Thus the nLab has
http://ncatlab.org/nlab/show/indiscrete+category as its own page.

>>Non evil is essentially evil.
>>I rather like this conclusion, don't you?

It is beautiful, but is it accurate?

>I'd expect the people abhoring evilness would
>say that full and faithful and essentially surjective is an "evil" notion
>of equivalence as opposed to the "good" one of adjoint pair where unit and
>counit are isos. The latter makes sense in any 2-category whereas the former
>doesn't. However, often you just get the "evil" version when not having
>a strong form of AC (for classes) available.

On the contrary, an ff and eso functor between two categories
is enough for the people who abhor evil, as far as I know,
to decide that the categories are equivalent (and so essentially the same).
Yet at the same time, these people tend to abhor AC!  How can this be?
It works if one works in a 2-category whose 1-morphisms are anafunctors.
Then it is a theorem requiring no choice (and true internal to any topos)
that any ff and eso functor can be enriched to an adjoint equivalence
(and in an essentially unique way).

Of course, "abhor" here should really be read as "consider optional".
It is possible to work with strict categories, or to work with AC,
but the main principles and results of category theory do not require either.


--Toby


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