Dear Jean,
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".
Non evil is essentially evil. I rather like this conclusion, don't you?
Of course, that's brilliant dialectics! 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. That's why your dialectics definitely applies! Best regards, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Thomas wrote:
The latter makes sense in any 2-category whereas the former doesn't.
One can call a 1-arrow in a 2-category ff (following Street) if it is representably so. Only the essential surjectivity causes problems, as 'surjectivity' is in general relative to some subcanonical pretopology on the ambient category. In a paper in preparation I have shown that any 2-category with a class J of ff 1-arrows which form a (strict) subcanonical pretopology can be localised (as a bicategory) at J, and we recover all the usual theorems about internal categories, groupoids etc. No assumptions about limits or colimits past what is necessary for the definition of J, so there are applications to various 2-categories of structured Lie groupoids (poss. infinite dimensional) and so forth to nicer-behaved categories. (I presented some of this at the Australian Category Seminar in 2011.)
However, often you just get the "evil" version when not having a strong form of AC (for classes) available
Indeed, and then you call them 'weak equivalences', and find some sort of model structure, or just localise outright. Also, for the purposes of further discussion, the terminology 'evil' has been demoted to a mere footnote at the nLab, as it was probably always meant to be by its coiners, and replaced by the morally neutral and more informative name 'principle of equivalence'. Interestingly, Voevodsky's Univalence Axiom is a way of ensuring the principle is always respected (and without awkward acrobatics). Best regards, David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I apologize for poor taste, but I do like chaotic: one might substitute indiscrete in the title ``Chaotic categories and equivariant classifying spaces'' (posted at http://front.math.ucdavis.edu/1201.5178), but surely not essentially subterminal. The comment I'd really like to make is that such categories can be seriously interesting. Peter On 4/29/13 3:05 PM, Toby Bartels wrote:
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
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 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/ ]
Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:
... I'd call it "essentially subterminal".
Hmm ... hitting a translation engine in a particularly good mood, I found "essentially terminal" rendering, in German, as "wesentlich unheilbar". (Round-tripping from there, you get "fundamentally incurable". Like that? Alas, it drew a blank on the actual proposal, "essentially subterminal" :-) .) Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:
... I'd call it "essentially subterminal".
Hmm ... hitting a translation engine in a particularly good mood, I found "essentially terminal" rendering, in German, as "wesentlich unheilbar". (Round-tripping from there, you get "fundamentally incurable". Like that? Alas, it drew a blank on the actual proposal, "essentially subterminal" :-) .) Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 02/05/2013 12:57 AM, Fred E.J. Linton wrote:
Thomas Streicher <streicher@mathematik.tu-darmstadt.de> suggested:
... I'd call it "essentially subterminal".
Hmm ... hitting a translation engine in a particularly good mood, I found "essentially terminal" rendering, in German, as "wesentlich unheilbar".
(Round-tripping from there, you get "fundamentally incurable". Like that? Alas, it drew a blank on the actual proposal, "essentially subterminal" :-)
Subterminal? Um, that would be "Unterseebootendbahnhof"? <grin, duck, & run> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
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David Roberts -
Fred E.J. Linton -
Peter May -
Robert Dawson -
Thomas Streicher -
Toby Bartels