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* Terminology
@ 2000-12-12  8:19 Jean Benabou
  0 siblings, 0 replies; 64+ messages in thread
From: Jean Benabou @ 2000-12-12  8:19 UTC (permalink / raw)
  To: Category list

I am confronted with problems of "contradictory terminology" which I would
like to solve and, since english is not my language, I need some
suggestions.
Let  F: Y-----> X be a functor such that for every object  x of  X the comma
category  (x,F) is connected.Such functors, although they are not defined in
all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory.
 The "cofinal" name comes obviously from the vocabulary of ordered sets
which are special cases, but in category theory  "co" is now associated with
dual notions.
The "initial" name is even less satisfactory, because:
(i) If  Y=1, F is identified with an object  x of X and F is "initial" iff x
is a terminal object of X  !
(ii) More generally, if Y has a terminal object  t  then F is "initial" iff 
F(t) is terminal !
(iii) Even more generally yet, without assuming the existence of terminal
objects in Y or X :
 Let X^ and Y^ be the categories of presheaves on X and Y, and  F! :X^----->
Y^  the canonical extension of F to these categories.If  T is the terminal
object of Y^ one can easily show that  F has the previous property iff 
F!(T) is terminal in X^.(Which by the way, gives the nicest proof of the
stability under composition of such functors)
I propose to call these functors either "terminal" or better "final" but I
would like to know if this would not conflict with previous terminology.
Thanks for your help. 



^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: Terminology
@ 2017-02-11 20:42 Fred E.J. Linton
  2017-02-14  8:48 ` Terminology Steve Vickers
       [not found] ` <02568D97-0A72-4CA8-8900-BDE11E890890@cs.bham.ac.uk>
  0 siblings, 2 replies; 64+ messages in thread
From: Fred E.J. Linton @ 2017-02-11 20:42 UTC (permalink / raw)
  To: Steve Vickers, Jean Benabou; +Cc: Categories

Steve, et al.,

If you want 

> a definition of "fish", but on the understanding that it has to include
whales

let me offer: "legless marine vertebrates" :-) .

Cheers, -- tlvp



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^ permalink raw reply	[flat|nested] 64+ messages in thread
* Terminology
@ 2017-02-09 22:03 Andrée Ehresmann
  0 siblings, 0 replies; 64+ messages in thread
From: Andrée Ehresmann @ 2017-02-09 22:03 UTC (permalink / raw)
  To: Categories

For Charles Ehresmann, the answer to Jean's question was that p be a
"homomorphism functor", a notion he already defined in his 1957 paper
"Gattungen in Lokalen Strukturen", reprinted in

   http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_I-1_et_I-2.pdf

In modern terms it should correspond to a faithful and amnestic functor.

Cordially
Andree



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^ permalink raw reply	[flat|nested] 64+ messages in thread
* Terminology
@ 2017-02-08  8:03 Jean Benabou
  2017-02-08 16:34 ` Terminology Jirí Adámek
                   ` (3 more replies)
  0 siblings, 4 replies; 64+ messages in thread
From: Jean Benabou @ 2017-02-08  8:03 UTC (permalink / raw)
  To: Categories

Dear all,

I'm sure the following question has been answered to. Could anyone
give me a precise answer and references to this answer. Many thanks.

QUESTION
Let  p: S --> X  be a functor. What conditions should satisfy p to be
called a structure functor, i.e. such that every object  s of S can be
thought of as a structure on the object  p(s).



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^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: Terminology
@ 2013-05-02  3:57 Fred E.J. Linton
  0 siblings, 0 replies; 64+ messages in thread
From: Fred E.J. Linton @ 2013-05-02  3:57 UTC (permalink / raw)
  To: Thomas Streicher , Jean Bénabou ; +Cc: Categories

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



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^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: Terminology
@ 2013-05-02  3:57 Fred E.J. Linton
  2013-05-03 11:53 ` Terminology Robert Dawson
  0 siblings, 1 reply; 64+ messages in thread
From: Fred E.J. Linton @ 2013-05-02  3:57 UTC (permalink / raw)
  To: Categories

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



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^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: Terminology
@ 2013-04-30  1:20 Fred E.J. Linton
  0 siblings, 0 replies; 64+ messages in thread
From: Fred E.J. Linton @ 2013-04-30  1:20 UTC (permalink / raw)
  To: Categories

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
> 


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* Terminology
@ 2013-04-24 17:13 Jean Bénabou
  2013-04-24 23:04 ` Terminology David Roberts
                   ` (3 more replies)
  0 siblings, 4 replies; 64+ messages in thread
From: Jean Bénabou @ 2013-04-24 17:13 UTC (permalink / raw)
  To: Categories

Dear all,

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?

Best regards,
Jean

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* Re: terminology
@ 2010-09-29  2:03 Todd Trimble
  0 siblings, 0 replies; 64+ messages in thread
From: Todd Trimble @ 2010-09-29  2:03 UTC (permalink / raw)
  To: Eduardo J. Dubuc; +Cc: Categories list


In reference to Eduardo's recent comment

>I feel the need to clarify some of my postings.
>
> Due to some public and private mails I realized that most people though
> that I
> was talking about the nLab.
> Well, all the time, when referring to "ghetto" or "subculture" I was
> aiming to
> the WHOLE of the category community within mathematics, not at the nLab
> within
> the category community.
>

I for one didn't get the impression you were referring to the
nLab.  My own comment was in response to Andre Joyal who
wrote "The 'evil' terminology is promoted by a small group of
peoples active in the nLab. It does not reflect a commun usage
in themathematical community."

I thought this could lead to a misunderstanding about the nLab,
hence my comment.

> Actually, I was not even aware of the existence of the nLab.
Due to this
> controversy, I visit the nLab and at first sight I essentially (not fully)
> agree with Andre's comments about the nLab in his recent posting.
>
> I say, go ahead !, nice work !
>
> I can add that I liked the lack of solemnity and the freedom to write down
> your understanding without fear to be wrong, and the freedom of the reader
> to
> insert comments and ask questions. The whole thing is very useful to all
> interested in the subjects being written about, and should not to be taken
> as
> a book in final form, which is not intended to be. Encyclopedia (18
> century)
> and Bourbaki are very important, but some fresh air is also important.
>

Thank you for the nice words (and I'm glad that -- even if nothing else
gets resolved -- at least this discussion has heightened awareness of the
existence of this project!).

The nLab (and the companion discussion forum, the nForum) are still
young and small.  It's a wiki, like Wikipedia, so that anyone can edit it.
Therefore, if you or anyone else sees flaws in an nLab article, you have
a warm open invitation to improve it!  It's easy to edit, and we appreciate
your leaving a note at the nForum to mention changes you make, or to
discuss anything you like.

We are a loosely aligned group with perhaps a dozen or so very active
contributors, including Andrew Stacey, Urs Schreiber, Zoran Skoda,
Mike Shulman, Toby Bartels, David Roberts, Tim Porter, David Corfield,
and myself. Perhaps the only things that really unite us are a belief in the
value of category theory and higher category theory, and a belief in the
value of this project.  Of course there is also Andre Joyal's CatLab,
which runs on the same easy-to-use and highly effective software.

Todd



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* terminology
@ 2010-09-28  4:38 Eduardo J. Dubuc
  0 siblings, 0 replies; 64+ messages in thread
From: Eduardo J. Dubuc @ 2010-09-28  4:38 UTC (permalink / raw)
  To: Categories list

I feel the need to clarify some of my postings.

Due to some public and private mails I realized that most people though that I
was talking about the nLab.
Well, all the time, when referring to "ghetto" or "subculture" I was aiming to
the WHOLE of the category community within mathematics, not at the nLab within
the category community.

Actually, I was not even aware of the existence of the nLab. Due to this
controversy, I visit the nLab and at first sight I essentially (not fully)
agree with Andre's comments about the nLab in his recent posting.

I say, go ahead !, nice work !

I can add that I liked the lack of solemnity and the freedom to write down
your understanding without fear to be wrong, and the freedom of the reader to
insert comments and ask questions. The whole thing is very useful to all
interested in the subjects being written about, and should not to be taken as
a book in final form, which is not intended to be. Encyclopedia (18 century)
and Bourbaki are very important, but some fresh air is also important.

I do not appreciate that a controversy about terminology be dismissed by
derision by saying

"thanks for trying to move the discussion away from terminology and back to
actual mathematical matters".

This kind of solemnity makes me shit !!

No need to move away from terminology, nobody is forbidding you to discuss
mathematics by discussing terminology, it is not one thing or the other.

We were talking about terminology, yes !!. Why not !. Terminology is
important, great mathematicians worried about it.

The "evil terminology" is wrong, somebody would even say evil, and it is
important that it does not establish itself.

This is not a fight, to abandon a terminology does not mean to loose a fight,
it just mean to become aware of some sides that were not properly considered
at the beginning. The looser is at the end the winner.

The challenge (not a minor challenge) is to find a good word "x" (or xxxx,
which means the same thing in spite to have four x's, ja!) to mean "invariant
under equivalence", or its negation, once we agree that such a word is
necessary due to the need of brevity justified by frequent use (if this
happens to be the case).

We can discuss the mathematics involved in the presence or lack of invariance
under equivalence, nobody forbids this by talking about the terminology utilized.

e.d.


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* Re: terminology
@ 2010-05-27 18:31 Colin McLarty
  0 siblings, 0 replies; 64+ messages in thread
From: Colin McLarty @ 2010-05-27 18:31 UTC (permalink / raw)
  To: categories

Yes, Michael has said in several papers that his foundation would be
given in FOLDS.  And perhaps some specific version of the axioms has
been given in some paper that I have missed.  But I do not know of it;
and when Martin-Löf suggested last year that I might want to pursue a
type-theoretic foundation for category theory he did not mention
knowing of one existing yet.  So far as I know it remains a project.

best, Colin



2010/5/26 John Baez <baez@math.ucr.edu>:
> Colin wrote:
>
>
> As to articulating a way to avoid ever using identity of objects and
>> identity of categories, John Baez writes
>>
>>> I think Michael Makkai has done it.  He has formulated a foundational
>>> approach to mathematics based on infinity-categories, in which equality
>>> plays no fundamental role:
>>>
>>> http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf
>>>
>>> I think some approach along these general lines might ultimately become
>> quite popular.
>>
>> But so far as  know, this remains an approach, and not any specific set of
>> axioms offered as foundation.
>>
>
>
> I should let Michael speak for himself, but I have the impression that he
> intends to found all his work on FOLDS - "first-order logic with dependent
> sorts".  In this paper:
>
> http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf
>
> he writes:
>
> "The restriction on the use of equality in FOLDS is a fundamental feature.
> FOLDS is to be used in formulating categorical situations in which, for
> example, equality of objects of a category is not an admissible primitive.
> The absence of term-forming operators, to be interpreted as
> functions, is a consequence of the absence of equality; it seems to me that
> the notion of "function" is incoherent without equality.
>
> It is convenient to regard FOLDS a logic without equality entirely, and deal
> with equality, as much as is needed of it, as extralogical primitives."
>
> Best,
> jb
>


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* Re terminology:
@ 2010-05-19 10:38 Ronnie Brown
  2010-05-20  7:58 ` soloviev
                   ` (3 more replies)
  0 siblings, 4 replies; 64+ messages in thread
From: Ronnie Brown @ 2010-05-19 10:38 UTC (permalink / raw)
  To: categories

Peter Sellinger writes recently: 

---------------------------------------------------

I think this is a very apt illustration of what happens if a term with
an existing meaning is redefined to mean something else. Henceforth it
is impossible for anybody to use the term (with either meaning)
without first giving a definition.

---------------------------------------------------

I completely agree. My own problem is with term `infinity groupoid' which  is used to describe something which is not even a groupoid, and whose use seems to me to militate against the understanding of what has been achieved with the original and much earlier definition. I once asked Gian-Carl Rota about such change of terminology, in connection with a refereeing job, and he agreed that mathematicians are used to creating confusion in this way. 

There are two easy tendencies: one is to use an old name in a quite different way, and the other is to use a new name for an old idea, so that the  use of the old term looks old fashioned, and a lot of work may be consigned to the dustbin of history, becoming not easy of access  for new students. 

It seems to be an example of these confusions is the way the simplicial singular complex of a space is called an infinity-groupoid, even the `fundamental infinity groupoid', when what seems to be referred to is that it is a Kan complex, i.e. satisfies the Kan extension condition, studied since 1955. The new term sounds like `dressing up' an old idea to look new. My personal objection to this change of terminology (i.e. axe to grind!) is that this distracts from studying the not so simple proofs that strict  higher homotopical structures exist, which mainly are for structured spaces (in particular filtered spaces (Brown/Higgins, Ashley), n-cubes of spaces (Loday), and more recently smooth spaces (Faria Martins/Picken)). The analysis and comparison of these uses should be made. It was certainly a relief to Philip and I that we could do something with filtered spaces which we could not do for the absolute case; the significance of the fact  that these constructions work and lead to specific calculations should be thought about. 

The term `higher dimensional group theory' which was published in a paper  with that title in 1982 was intended to suggest developing higher groupoid theory and its relations to homotopy theory in the spirit of group theory, which meant specific constructions relevant to geometry and calculations, even computer calculations,  of many examples, in which actual numbers arise as a test of and examples of the general theory, and in which some aspects of group theory are sensibly seen as better represented in the higher dimensional theory; and example of this is the nonabelian tensor  product of groups, where group theorists have found lots of pickings. 

I am not sure how these terminological problems will be resolved, and I know the term (\infty,n)-groupoid has been well used recently but the problem of relation to the older ideas, which have had a certain success, should be recognised. 

Ronnie Brown








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* Re: terminology
@ 2010-05-16 12:44 Peter Selinger
  0 siblings, 0 replies; 64+ messages in thread
From: Peter Selinger @ 2010-05-16 12:44 UTC (permalink / raw)
  To: Categories List

I had written:
> 
> My last comment is that, unlike what Jeff Egger claimed, "autonomous
> category" is not a special case of "*-autonomous category", because no
> symmetry is assumed in autonomous categories. Unless of course one
> first drops symmetry from the definition of *-autonomous categories,
> as Jeff has also suggested. As it stands, neither of "autonomous" and
> "*-autonomous" implies the other, which is perfectly fine in my
> opinion, since they are two different words.

I would like to clarify that Jeff himself did not say anything false,
because in the context in which he said it, he had in fact assumed the
non-symmetric definition of *-autonomous category (of [Barr 1995]).
Sorry if it sounded like I was accusing him.

My intention was only to point out that the statement "autonomous
categories are a special case of *-autonomous categories" cannot be
quoted out of context, because it is false under the original
definition of *-autonomous category that includes symmetry (of [Barr
1979]). Since it had already been quoted out of context when I wrote
the above, I just wanted to point out how the potential confusion. 

I think this is a very apt illustration of what happens if a term with
an existing meaning is redefined to mean something else. Henceforth it
is impossible for anybody to use the term (with either meaning)
without first giving a definition. That's no problem in a math paper,
where definitions are usually given or cited anyway, and therefore
terminology is in principle arbitrary. But it does tend to hobble
everyday discussion.

-- Peter

P.S.: since I have a demonstrated ability to put my foot in my mouth,
I'd like to clarify that I am not accusing Mike Barr of anything
either. His 1995 paper is clearly entitled "Non-symmetric *-autonomous
categories", and the inside of the paper clearly explains the
distinction. It is only in subsequent use that any confusion arises.
The usual solution, of putting either (non-symmetric) or (symmetric)
in parentheses the first time the term is used, and omitting it for
subsequent uses, is perfectly adequate. I am very happy with the
statement "an autonomous category is a special case of a
(non-symmetric) *-autonomous category".

M. Barr (1979). "*-Autonomous Categories", Lectures Notes in
Mathematics 752. Springer. 

M. Barr (1995). "Non-symmetric *-autonomous categories". 
Theoretical Computer Science 139:115–130.


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^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: bilax_monoidal_functors
@ 2010-05-13 17:17 Michael Shulman
  2010-05-14 14:43 ` terminology (was: bilax_monoidal_functors) Peter Selinger
  0 siblings, 1 reply; 64+ messages in thread
From: Michael Shulman @ 2010-05-13 17:17 UTC (permalink / raw)
  To: Jeff Egger

On Mon, May 10, 2010 at 2:28 PM, Jeff Egger <jeffegger@yahoo.ca> wrote:
> the fact that "autonomous category" is a special case (and, from one
> point of view, a rather uninteresting special case) of
> "star-autonomous category", whereas it sounds like "star-autonomous
> category" should mean an "autonomous category" with some extra
> structure.

I agree, it does sound like that, but there is at least a long
tradition of such names in mathematics (not that that makes
them a good thing).
(http://ncatlab.org/nlab/show/red+herring+principle)

One reason I like "autonomous" to mean a symmetric monoidal category
in which all objects have duals is that the only alternative names I
have heard for such a thing convey misleading intuition to me.  They
are sometimes called "compact closed" or (I think) "rigid" monoidal
categories, but "compact" and "rigid" are words with definite and
inapplicable intuitive meanings for me.  Compact means small, finite,
bounded, inaccessible by directed joins, etc. and "rigid" means "having few
automorphisms," and I don't see that there is anything very compact or
rigid about such categories.  The only relationship I can think of is that a
compact subset of a Hausdorff space is closed, and a symmetric monoidal
category with duals for objects is also automatically closed, but of course
these two meanings of "closed" are totally different.  Perhaps someone
can enlighten me?

Mike


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^ permalink raw reply	[flat|nested] 64+ messages in thread
* RE : bilax monoidal functors
@ 2010-05-08  3:27 John Baez
  2010-05-10 18:16 ` bilax_monoidal_functors?= John Baez
  0 siblings, 1 reply; 64+ messages in thread
From: John Baez @ 2010-05-08  3:27 UTC (permalink / raw)
  To: categories

André Joyal wrote:

I am using the following terminology for
> higher braided monoidal (higher) categories:
>
> Monoidal< braided < 2-braided <.......<symmetric
>
> A (n+1)-braided n-category is symmetric
> according to your stabilisation hypothesis.
>
> Is this a good terminology?
>

I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided".  This
seems preferable to me, not because it sounds nicer - it doesn't - but
because it starts counting at a somewhat more natural place.  I believe that
counting monoidal structures is more natural than counting braidings.

For example, a doubly monoidal n-category, one with two compatible monoidal
structures, is a braided monoidal n-category.    I believe this is a theorem
proved by you and Ross when n = 1.  This way of thinking clarifies the
relation between braided monoidal categories and double loop spaces.

Various numbers become more complicated when one counts braidings rather
than monoidal structures:

An n-tuply monoidal k-category is (conjecturally) a special sort of
(n+k)-category... while an n-braided category is a special sort of
(n+k+1)-category.

Similarly: n-dimensional surfaces in (n+k)-dimensional space are n-morphisms
in a k-tuply monoidal n-category... but they are n-morphisms in an
(k-1)-braided n-category.

And so on.

On the other hand, if it's braidings that you really want to count, rather
than monoidal structures, your terminology is perfect.

By the way: I don't remember anyone on this mailing list ever asking if
their own terminology is good.  I only remember them complaining about other
people's terminology.  I applaud your departure from this unpleasant
tradition!

Best,
jb


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^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: terminology
@ 2007-01-27 17:06 wlawvere
  0 siblings, 0 replies; 64+ messages in thread
From: wlawvere @ 2007-01-27 17:06 UTC (permalink / raw)
  To: categories

Dear Eduardo and everybody:

In one of your papers you used the term
Nullstellensatz for a special case (in some
sense an "algebraically closed"case). I
propose to use that term in this more
general case.

The parameters for various
traditional cases can be perhaps expessed
by an essential connected morphism of
toposes E->S. That is, a full inclusion of
"relatively discrete" into "relatively
 continuous" which has both left adjoint
("connected components") and right adjoint
("points").

In that context there is a natural map from
points to components; if it is epic, we can
say that the Nullstellensatz holds for
E->S.

If S is just the category of abstract sets,
one could think of E as algebraically closed if
the Nullstellensatz holds.

But as seems implicit in Galois theory, for
algebraic geometry over a non-algebraically
closed K, the appropriate base topos S consists
not of abstract sets, but rather of sheaves
on C = the opposite of the category of finite
extensions of K, with every map covering. If E
is the topos of sheaves on (finitely generated
K-algebras )^op with respect to a topology that
restricts to the above on C, I believe
we have a classical example of both your
formulation and mine.

Bill

PS There are other stronger results that also
could be called  Nullstellensatz, involving
another topos F between E and S, such as
the one generated by algebras that are finite
dimensional as K-vector spaces, or one
suggested by Birkhoff's SDI theorem. What
is the appropriate statement for these results ?


Quoting Eduardo Dubuc <edubuc@dm.uba.ar>:

> hello:
>
> Given a set CC of objects in a topos EE, consider the following
> property:
>
>       " X no= empty  iff  exists C \in CC, hom(C, X) no= empty "
>
> example; CC = a set of generators
>
> Has (this property) already  a name ?
>
> If not, can you suggest one ?
>
> Any answer will be welcome.
>
> (Notice that if CC is a set of points (instead of objects) we say
> that
> there are enough points)
>
> Thanks          Eduardo J. Dubuc
>
>
>
>




^ permalink raw reply	[flat|nested] 64+ messages in thread
* terminology
@ 2007-01-26 23:30 Eduardo Dubuc
  0 siblings, 0 replies; 64+ messages in thread
From: Eduardo Dubuc @ 2007-01-26 23:30 UTC (permalink / raw)
  To: categories

hello:

Given a set CC of objects in a topos EE, consider the following property:

      " X no= empty  iff  exists C \in CC, hom(C, X) no= empty "

example; CC = a set of generators

Has (this property) already  a name ?

If not, can you suggest one ?

Any answer will be welcome.

(Notice that if CC is a set of points (instead of objects) we say that
there are enough points)

Thanks          Eduardo J. Dubuc




^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re:  terminology
@ 2005-12-30  1:16 vs27
  0 siblings, 0 replies; 64+ messages in thread
From: vs27 @ 2005-12-30  1:16 UTC (permalink / raw)
  To: Categories

On Dec 29 2005, Vaughan Pratt wrote:

> Without taking sides on the prone/supine terminology question, I do have
> a strong reaction to the Benabou/May/Dubuc concern that respect for a
> field is undermined by its adoption of frivolous terminology.
>
Dear Vaughan, as everybody has a say. Just my views.
I prefer some nomenclature that sounds mathematical,
rather than based on the name of a friend or a private joke.
(may be i don't understand all the jokes ?)
Also in any case one should avoid  renaming existing
concepts, that is just not fair.


Good opportunity to wish happy new year to everybody.








^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re:  terminology
@ 2005-12-29 19:09 Nikita Danilov
  0 siblings, 0 replies; 64+ messages in thread
From: Nikita Danilov @ 2005-12-29 19:09 UTC (permalink / raw)
  To: Categories

Vaughan Pratt writes:
 > Without taking sides on the prone/supine terminology question, I do have
 > a strong reaction to the Benabou/May/Dubuc concern that respect for a
 > field is undermined by its adoption of frivolous terminology.
 >
 > This may be a valid concern for a young field like category theory, but
 > for a more mature subject such as physics, a more relevant concern is
 > the undermining of the ability to poke fun at oneself by the fear of not
 > being taken seriously.
 >
 > Has the adoption of frivolous nomenclature for quarks ("strange,"
 > "charm," "beauty" and even "quark" itself) diminished in any way the
 > world's respect for quarks and their investigators?

There indeed are drawbacks whenever scientific terms are contrary to the
centuries old tradition not taken from Greek or Latin languages (that,
thanks to their very regular and flexible system of word formation are
so suitable for taxonomies) shared by many cultures. For one thing,
words of existing languages are not in one to one mapping, and then a
term from contemporary language may be not culturally neutral (consider
silly naming wars for transuranium elements).

On the other hand, I stopped using "co-product" after more than one
person with the background in classical languages read it as
"copro-duct".

 >
 > And what of computational topology?  Should we turn a blind eye to
 > whether Scott is sober, and substitute a more genteel euphemism for his
 > bottom?
 >
 > Vaughan Pratt

Nikita.




^ permalink raw reply	[flat|nested] 64+ messages in thread
* Terminology
@ 2005-12-10  3:51 jean benabou
  2005-12-21 20:04 ` Terminology Eduardo Dubuc
  0 siblings, 1 reply; 64+ messages in thread
From: jean benabou @ 2005-12-10  3:51 UTC (permalink / raw)
  To: Categories

I have seen in this mail that the suggestion of Taylor and Johnstone to
replace cartesian and cocartesian maps by prone and supine ones begins to
be accepted. When I first saw that suggestion, I was so amazed that I
thought it was a joke, and not such a good one. I still hope it is no more
than that. But, just in case, and before it is too late, I want to say
that I am very strongly opposed to such changes for many reasons:
linguistic, mathematical, and ethical, which I am ready to explain in
detail if I am asked to do so.





^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: terminology
@ 2003-10-17 15:19 Marco Grandis
  0 siblings, 0 replies; 64+ messages in thread
From: Marco Grandis @ 2003-10-17 15:19 UTC (permalink / raw)
  To: categories

In reply to Stasheff's question on terminology for homotopy coherent algebras:

>but now what about e.g. 1-homotopy associaitve satisfying a STRICT
>pentagon??
>perhaps strict 1-homotopy

I would say:

"2-strict sha-algebra", as motivated below.
(sha = strongly homotopy associative)

However: after the strict pentagon, this structure has a second coherence
condition for the associativity homotopy
(which disappears for monoidal categories, just because their 2-morphisms
are trivial)

____________

In a paper [*] on strongly homotopy associative (differential) algebras, I
proposed this definition (4.2; pages 38-39).

Notation: a sha-algebra is a graded module  A  with morphisms
(sort of components of a global differential  d of bar coalgebras)

   d_1: A --> A  (degree - 1; the differential)

   d_2: AoA --> A  (degree 0; the product)

   d_3: AoAoA --> A  (degree 1; the associativity 1-homotopy)
   ........
   d_n: A^n  -->  A   (degree  n - 2; the coherence n-homotopy)
   ........

( o = tensor product;  ^n  = tensor power)

under axioms

(1)  d_1.d_1 = 0
(2)  ....

(expressing  dd = 0  for the global differential).

DEF. This is called an *n-strict sha-algebra* if  d_p = 0  for  p > n.

Equivalently,  the morphisms  d_1,..., d_n  have to satisfy the original
axioms (1)  ... (n)
plus n - 1 conditions obtained from the axioms  (n+1) ... (2n - 1),
cancelling the null  d_p's
(the remaining axioms become trivial).

This gives:

1-strict = differential module

2-strict = associative differential algebra

3-strict = 1-homotopy associative differential algebra
   with strict pentagon (from axiom (3)) and axiom (4) reduced to:

   (4)   d3 (1o1od3 + 1od3o1 + d3o1o1) = 0.

_______

So far in that paper.
The name is chosen to make  d_n  the last relevant component, in the
n-strict case.

I might now (more geometrically) prefer a - 1 shift in these names, so that
the last example would be named 2-strict, in accord with the fact that the
last relevant homotopy is a an ordinary ("one-dimensional") homotopy and
everything becomes strict starting with "dimension 2".

_______

Reference:

[*] M. Grandis, On the homotopy structure of strongly homotopy associative
algebras, J. Pure Appl. Algebra 134 (1999), 15-81.

_______

Regards    MG







^ permalink raw reply	[flat|nested] 64+ messages in thread
* terminology
@ 2003-10-16 21:39 James Stasheff
  0 siblings, 0 replies; 64+ messages in thread
From: James Stasheff @ 2003-10-16 21:39 UTC (permalink / raw)
  To: dmd1, categories

In `higher homotopy theory', terminology has not setled down nor is it
transparent

homotopy ___________ algebra can mean a variety of things

letting ______________ = associative
it can mean JUST that there is a homtopy for associaitivity
or
some authors use it to mean A_\infty

which I initially tried to indicate by strongly homtopy associative

_\infty seems to have caught on to mean the presence of higher homtopies
of all orders

in most but not all cases, such algebras have a homtopy invariant
defintion

so I would suggest the following revisionist terminology

1-homotopy associative means JUST that there is a homotopy for
associaitivity

similarly n-homotopy associative would mean homotopies of homotopies
of...

homotopy invariant ___ algebra would mean just what it says

so far so good
but now what about e.g. 1-homotopy associaitve satisfying a STRICT
pentagon??

perhaps strict 1-homotopy

open to suggestions



	Jim Stasheff		jds@math.upenn.edu

		Home page: www.math.unc.edu/Faculty/jds

As of July 1, 2002, I am Professor Emeritus at UNC and
I will be visiting U Penn but for hard copy
        the relevant address is:
        146 Woodland Dr
        Lansdale PA 19446       (215)822-6707





^ permalink raw reply	[flat|nested] 64+ messages in thread
* Terminology
@ 2001-04-09 11:06 Krzysztof Worytkiewicz
  0 siblings, 0 replies; 64+ messages in thread
From: Krzysztof Worytkiewicz @ 2001-04-09 11:06 UTC (permalink / raw)
  To: categories

Dear Categories,

Is it  established terminology  to call *injective* a faithful functor
which is injective (in the usual sense) on objects ?

Cheers, Krzysztof




^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: Terminology
@ 2000-12-14  6:17 Max Kelly
  0 siblings, 0 replies; 64+ messages in thread
From: Max Kelly @ 2000-12-14  6:17 UTC (permalink / raw)
  To: categories

In response to Jean Benabou's question about the terminology for what some
call "cofinal" functors, may I refer him to Section 4.5 of my book "Basic
Concepts of Enriched Category Theory", where such notions are considered
in considerable generality? In so far as we deal with functors - meaning
"V-functors" in the context of V-enriched category theory - the terms I
used, which are those common here at Sydney, are "final functor" and
"initial functor". These notions, however, make sense only when V is
cartesian closed; for a more general symmetric monoidal closed V, what is
said to be initial is a pair (K,x) where K is a V-functor A --> C and x is
a V-natural transformation H --> FK, where H: A --> V and F: C --> V are
V-functors with codomain V, and thus are "weights" for weighted limits.
The 2-cell x expresses F as the left Kan extension of H along K if and
only if, for every V-functor T: C --> B of domain C, the canonical
comparison functor (induced by K and x) between the weighted limits, of
the form

                   (K,x)* : {F,T} ----> {H,TK},
		   
is invertible (either side existing if the other does); the book contains
a third equivalent form making sense whether the limits exist or not. When
these equivalent properties hold, the pair (K,x) is said to be INITIAL.
The point is that, in this case, the F-weighted limit of any T can be
calculated as the H-weighted limit of TK.

When V is cartesian closed, we have for each V-category C the V-functor C
---> V constant at the object 1, limits weighted by which are the CONICAL
limits, which when V = Set are the classical limits. For such a V we can
consider the special case of the situation considered above, where each of
H and F is the functor constant at the object 1, and where x is the unique
2-cell between H and FK; we call the functor K "initial" when this pair
(K,x) is so; equivalently when the canonical lim T ---> lim TK is
invertible for every T (for which one side exists -- or better put in
terms of cones), or equivalently again when

           colim C(K-,c) == 1  for each object c of C.
	  
When V = Set, this is just to say that each comma-category K/c is
connected. When the category C is filtered, a fully-faithful K: A --> C is
final (dual to initial) precisely when each c/K is non-empty.

The book goes on to discuss the Street-Walters factorization of any (ordinary)
functor into an initial one followedby a discrete op-fibration.	

The above being so, it seems that Jean's good taste has led him to suggest
the very same nomenclature that recommended itself to us at Sydney.  I
should have been happier, though, if he had recalled the treatment I gave
lovingly those many years ago. There are many other expositions in the
book that I am equally happy with, and which I am sure Jean would enjoy.
By the way, someone spoke recently on this bulletin board of the book's
being out of print and hard to get; I've been meaning to find the time to
reply to that, and discuss what might be done. The copyright has reverted
to me; but the text does not exist in electronic form - it was written
before TEX existed, and prepared on an IBM typewriter by an excellent
secretary with nine balls.

I suppose I could have some copies - one or more hundreds - printed from
the old master, after correcting the observed typos. But the photocopying
and binding and the postage would cost a bit. I'ld be happy to receive
suggestions, especially from such colleagues as would like to get hold of
a copy. By the way, I sent out preprint copies to about 100 colleagues
back in 1980 or 1981; if any of those are still around, I point out that
they contain the full text. So too do those copies which appeared in the
Hagen Seminarberichte series. Once again, I look forward to any comments,
either in favour of or against making further copies.

Max Kelly.




^ permalink raw reply	[flat|nested] 64+ messages in thread
[parent not found: <3a35cdd73a39f901@amyris.wanadoo.fr>]
* Re: Terminology
@ 2000-12-13  1:17 Steve Lack
  0 siblings, 0 replies; 64+ messages in thread
From: Steve Lack @ 2000-12-13  1:17 UTC (permalink / raw)
  To: categories

Jean Benabou writes:
 > I am confronted with problems of "contradictory terminology" which I would
 > like to solve and, since english is not my language, I need some
 > suggestions.
 > Let  F: Y-----> X be a functor such that for every object  x of  X the comma
 > category  (x,F) is connected.Such functors, although they are not defined in
 > all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's
 > handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory.

Mac Lane calls such functors ``final'' in Categories for the Working
Mathematician. I do too.

Steve Lack.





^ permalink raw reply	[flat|nested] 64+ messages in thread
* re: terminology
@ 2000-01-28 12:02 James Stasheff
  0 siblings, 0 replies; 64+ messages in thread
From: James Stasheff @ 2000-01-28 12:02 UTC (permalink / raw)
  To: categories

So far the clear front runner is `face complex'
Thanks to all the nominators.

Grandis points out why the historical semi-simplicial
won't fly for at least another generation.

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds




^ permalink raw reply	[flat|nested] 64+ messages in thread
* Re: terminology
@ 2000-01-28  9:57 Marco Grandis
  0 siblings, 0 replies; 64+ messages in thread
From: Marco Grandis @ 2000-01-28  9:57 UTC (permalink / raw)
  To: categories, James Stasheff

J. Stasheff wrote:

>Has terminology settled down?
>I can recall seeing various terms for
>``simplicial object without degeneracies''


I am afraid it has not.

In my opinion, it should be called 'semi-simplicial object', consistently
with the original terminology in Eilenberg-Zilber (see references below).
Such a term has been adopted in Weibel's text on homological algebra
(1994). But there seems to be some opposition.
___

I hope the following reconstruction of terminology is correct.

1. What is now called a simplicial object was introduced by Eilenberg and
Zilber (1950); they use:

(a) [already existing] 'simplicial complex' = set with distinguished parts;
(b) [new term] 'semi-simplicial complex' = graded set with faces;
(c) [new term] 'complete s.s. complex' = graded set with faces and degeneracies;

2. Later, notion (c) was recognised as more important than (b) and called
'semi-simplicial complex', leaving (b) without any standard name.

3. Since May's book (1967) at least, notion (c) gradually settled down as
'simplicial set', generalised to 'simplicial object' in a category; this is
now standard.

4. It should now be natural to use a similar term, 'semi-simplicial object
(possibly: set)' for (b), i.e. a 'simplicial object without degeneracies'
(as in Weibel 1994). This is consistent with the original use in
Eilenberg-Zilber and gives a non-ambiguous set of terms for the three
notions recalled:

(a) 'simplicial complex' (also: combinatorial complex)
(b) 'semi-simplicial object (set)'
(c) 'simplicial object (set)'

However, I used myself this terminology in a paper published in '97 and had
strong reactions from people attached to the terminology in use between
50's and '60s (point 2 above).

___

References:

S. Eilenberg - J.A. Zilber, Semi-simplicial complexes and singular
homology, Ann. of Math. 51 (1950), 499-513.

J.P. May, Simplicial objects in algebraic topology, Van Nostrand 1967.

C.A. Weibel, An introduction to homological algebra, Cambridge Univ. Press,
Cambridge, 1994.

___

With best regards

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752

http://www.dima.unige.it/STAFF/GRANDIS/
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/






^ permalink raw reply	[flat|nested] 64+ messages in thread
* terminology
@ 2000-01-27 19:28 James Stasheff
  2000-01-27 21:04 ` terminology Paul Glenn
  0 siblings, 1 reply; 64+ messages in thread
From: James Stasheff @ 2000-01-27 19:28 UTC (permalink / raw)
  To: categories

Has terminology settled down?
I can recall seeing various terms for 
``simplicial object without degeneracies''

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds




^ permalink raw reply	[flat|nested] 64+ messages in thread
end of thread, other threads:[~2017-02-14  9:39 UTC | newest]

Thread overview: 64+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2000-12-12  8:19 Terminology Jean Benabou
  -- strict thread matches above, loose matches on Subject: below --
2017-02-11 20:42 Terminology Fred E.J. Linton
2017-02-14  8:48 ` Terminology Steve Vickers
     [not found] ` <02568D97-0A72-4CA8-8900-BDE11E890890@cs.bham.ac.uk>
2017-02-14  9:39   ` Terminology Jean Benabou
2017-02-09 22:03 Terminology Andrée Ehresmann
2017-02-08  8:03 Terminology Jean Benabou
2017-02-08 16:34 ` Terminology Jirí Adámek
2017-02-10  1:42   ` Terminology George Janelidze
2017-02-08 21:40 ` Terminology Carsten Führmann
2017-02-09 11:31 ` Terminology Thomas Streicher
     [not found] ` <20170208180636.18346065.28939.42961@rbccm.com>
2017-02-09 16:38   ` Terminology Jean Benabou
2017-02-11 15:07     ` Terminology Steve Vickers
2013-05-02  3:57 Terminology Fred E.J. Linton
2013-05-02  3:57 Terminology Fred E.J. Linton
2013-05-03 11:53 ` Terminology Robert Dawson
2013-04-30  1:20 Terminology Fred E.J. Linton
2013-04-24 17:13 Terminology Jean Bénabou
2013-04-24 23:04 ` Terminology David Roberts
2013-04-27 13:08 ` Terminology Thomas Streicher
     [not found] ` <20130427130857.GC16801@mathematik.tu-darmstadt.de>
2013-04-28  3:49   ` Terminology Jean Bénabou
2013-04-28 22:47     ` Terminology Olivier Gerard
     [not found] ` <557435A6-4568-4012-8C63-E031931F41FB@wanadoo.fr>
2013-04-28 14:17   ` Terminology Thomas Streicher
2013-04-29 20:05     ` Terminology Toby Bartels
2013-04-30  0:58       ` Terminology Peter May
2010-09-29  2:03 terminology Todd Trimble
2010-09-28  4:38 terminology Eduardo J. Dubuc
2010-05-27 18:31 terminology Colin McLarty
2010-05-19 10:38 Re terminology: Ronnie Brown
2010-05-20  7:58 ` soloviev
2010-05-20 19:53   ` terminology Eduardo J. Dubuc
     [not found] ` <AANLkTikre9x4Qikw0mqOl1qZs9DDSkcBu3CXWA05OTQT@mail.gmail.com>
2010-05-21 17:00   ` Re terminology: Ronnie Brown
     [not found]     ` <B3C24EA955FF0C4EA14658997CD3E25E370F5827@CAHIER.gst.uqam.ca>
2010-05-22 21:43       ` terminology Ronnie Brown
     [not found]       ` <4BF84FF3.7060806@btinternet.com>
2010-05-22 22:44         ` terminology Joyal, André
2010-05-23 15:39           ` terminology Colin McLarty
2010-05-24 18:04             ` terminology Vaughan Pratt
2010-05-26  3:08               ` terminology Toby Bartels
2010-05-25 14:08             ` terminology John Baez
2010-05-26  8:03             ` terminology Reinhard Boerger
2010-05-25 19:39 ` terminology Colin McLarty
2010-05-29 21:47   ` terminology Toby Bartels
2010-05-30 19:15     ` terminology Thorsten Altenkirch
     [not found]     ` <A46C7965-B4E7-42E6-AE97-6C1D930AC878@cs.nott.ac.uk>
2010-05-30 20:51       ` terminology Toby Bartels
2010-06-01  7:39         ` terminology Thorsten Altenkirch
2010-06-01 13:33           ` terminology Peter LeFanu Lumsdaine
     [not found]         ` <7BF50141-7775-4D3C-A4AF-D543891666B9@cs.nott.ac.uk>
2010-06-01 18:22           ` terminology Toby Bartels
     [not found] ` <AANLkTilG69hcX7ZV8zrLpQ_nf1pCmyktsnuE0RyJtQYF@mail.gmail.com>
2010-05-26  8:28   ` terminology John Baez
2010-05-16 12:44 terminology Peter Selinger
2010-05-13 17:17 bilax_monoidal_functors Michael Shulman
2010-05-14 14:43 ` terminology (was: bilax_monoidal_functors) Peter Selinger
2010-05-15 19:52   ` terminology Toby Bartels
2010-05-08  3:27 RE : bilax monoidal functors John Baez
2010-05-10 18:16 ` bilax_monoidal_functors?= John Baez
2010-05-11  8:28   ` bilax_monoidal_functors?= Michael Batanin
2010-05-12  3:02     ` bilax_monoidal_functors?= Toby Bartels
2010-05-13 23:09       ` bilax_monoidal_functors?= Michael Batanin
2010-05-15 16:05         ` terminology Joyal, André
2007-01-27 17:06 terminology wlawvere
2007-01-26 23:30 terminology Eduardo Dubuc
2005-12-30  1:16 terminology vs27
2005-12-29 19:09 terminology Nikita Danilov
2005-12-10  3:51 Terminology jean benabou
2005-12-21 20:04 ` Terminology Eduardo Dubuc
2005-12-26 19:47   ` terminology Vaughan Pratt
2005-12-29 23:17     ` terminology Eduardo Dubuc
2006-01-04 14:59       ` terminology Eduardo Dubuc
2003-10-17 15:19 terminology Marco Grandis
2003-10-16 21:39 terminology James Stasheff
2001-04-09 11:06 Terminology Krzysztof Worytkiewicz
2000-12-14  6:17 Terminology Max Kelly
     [not found] <3a35cdd73a39f901@amyris.wanadoo.fr>
2000-12-13 11:10 ` Terminology Dr. P.T. Johnstone
2000-12-13  1:17 Terminology Steve Lack
2000-01-28 12:02 terminology James Stasheff
2000-01-28  9:57 terminology Marco Grandis
2000-01-27 19:28 terminology James Stasheff
2000-01-27 21:04 ` terminology Paul Glenn

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