Re: Terminology

["Fred E.J. Linton" <fejlinton@usa.net>]

Forgive my repeating, perhaps unnecessarily, the obvious, but
without doing that I fear I may just get inextricably lost as I
try, once again, to sort my way through this question to more of
an answer than I was able to access the last times I tried.

If we pay momentary attention to the "underlying point-set" functor,
from the category of Topological Spaces to that of Sets, we see that
it "has" both a left adjoint, assigning to each set that self-same set
in its discrete topology, and a right adjoint, assigning to each set
that self-same set in its indiscrete topology.

That said, let me turn instead to the "underlying set of objects"
functor from that category of all small categories to that of sets.
It, too, has both a left adjoint, assigning to each set "the" category
having that self-same set as its set of objects, but admitting no 
morphisms between any two objects other than identity maps where 
identity maps are absolutely required -- what's known as the discrete
category on that set of objects -- and a right adjoint, assigning to
each set "the" category having that self-same set as its set of objects,
with the peculiar feature that each of its hom-sets has cardinality 1
-- category that, by analogy with the topological right adjoint, one
might (as Toby Bartels so deftly reminds us) choose to call indiscrete.

And if these categories are nothing more nor less than those that Jean 
Bénabou envisages, with functor to the terminal category 1 fully faithful,
then I guess "indiscrete" would be my answer, too, to his question,
"what would you call" such a category? But for me the indiscreteness
is not in any way a reflection of that full fidelity -- rather, it is
a reflection of the parallel between the fact that such a category "is" 
the value of the right adjoint to the "underlying set of objects" functor
and that an indiscrete space serves as value of the right adjoint to the 
"underlying point-set" functor.

"Setoïd"? "essentially subterminal"? Come on, folks, give us a break :-) !

Cheers, -- Fred

------ Original Message ------
Received: Mon, 29 Apr 2013 07:53:37 PM EDT
From: Toby Bartels <categories@TobyBartels.name>
To: Categories <categories@mta.ca>
Subject: categories: Re: Terminology

> Thomas Streicher wrote:
> 
>>Jean Bénabou 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
> 


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]