Re: co-

Re: co-

[John R Isbell]

   The Shorter OED is doubtless not an infallible
guide to mathematical etymology, but it has
something obvious that we have all been missing
(as far as I have seen yet):

    <2 {\it Math.} Short for {\it complement},
   in the sense 'of the complement' (as 
   {\it cosine}), or 'complement of' (as
   {\it co-latitude}).>

All categorical 'co's are surely that kind. In
prticular, cohomology like cosine. If Steenrod 
had in mind covariant, contravariant, why would 
he say 'cohomology'? Had he said 
'contrahomology' it would be clear. (It is
relevant, I think, that in Lefschetz' first
Colloquium book he called cohomology, such as
$H^1$, homology in negative dimensions, as
$H_{-1}$.)
   Cofinal has to be Latin 'cum'+final.
       John

Re: co-

[Robert Dawson]

----------
> From: Paul Taylor <pt@dcs.qmw.ac.uk>
> To: categories@mta.ca
> Subject: categories: co-
> Date: Friday, July 03, 1998 8:39 AM
> 
> What are the origins of the   co-   prefix, as in coproduct, coequaliser
..,
> and who established their use?
> 
> Has anybody ever thought through and written down any guidelines on
> which of a pair of dual concepts is co-?
> 
> Who is reponsible for dropping this prefix from cofinal?

	Njectural answer: anybody who doesn't want the ncept to be nfused
with "cofinal" in the topological sense, which is surely an older usage.
(Is this rrect?)

		Robert Dawson

RE: co-

[Fred E J Linton]

Hi, all,

Jurgen asks:

> Why are certain categorical notions preferred over their dual
> counterparts?  E.g., hardly anyone talks about the Yoneda embedding
> of A into [A,Set]^op.

One point of my old (alas still unpublished) remarks "Sur les choix de variance
predestinees" was exactly why one "should" only see those Yoneda maps in the
forms  A ---> [A,Set]^op  -- and  A ---> [A^op, Set]  -- but no others (!). 
[First Ehresmann conf., Paris/Fontainebleau, 197?.] 

-- Fred

RE: co-

[Vaughan Pratt]

From: Fred E J Linton <FEJLINTON/0004142427@MCIMAIL.COM>
>One point of my old (alas still unpublished) remarks "Sur les choix de variance
>predestinees" was exactly why one "should" only see those Yoneda maps in the
>forms  A ---> [A,Set]^op  -- and  A ---> [A^op, Set]  -- but no others (!). 
>[First Ehresmann conf., Paris/Fontainebleau, 197?.] 

Mildly apropos of this, the two maps can be rolled into one, in a sense,
to give the "bi-Yoneda embedding" F:C->Chu(Set,|C|) that I presented at
the Barrfest, where |C| denotes the set of arrows of C.

This embedding represents each object b of C as the Chu space F(b) =
(A,r,X), r:AxX->K, where A is the set of arrows f:a->b over all a,
X is the set of arrows h:b->c over all c, and r(f,h) = hf.

Each morphism g:b->b' is represented as the pair F(g) = (j,k) of functions
j:F_A(b)->F_A(b'), k:F_X(b')->F_X(b) defined by j(f) = gf, k(h) = hg
for each point f:a->b in F_A(b) and state h:b'->c in F_X(b').  (j,k) is
a Chu transform (= continuous, = satisfies the adjointness condition).

F is full, faithful, concrete with respect to U:C->Set defined by U(b)
{f:a->b} ("left" Yoneda), and co-concrete with respect to V:C->Set^op
defined by V(b) = {h:b->c} ("right" Yoneda) (or the other way round
depending on which way you're facing).

Regrettably this didn't go in the proceedings, being already committed
to TCS.

Vaughan Pratt

re: co-

[Koslowski]

Ross Street brought up an important point.  As you start considering
higher-dimensional categories, the number of possible dualizations
rises exponentially.  Also, given a monoidal category V, we can consider
it either as a bicategory with trivial hom-categories, or as a
one-object bicategory (usually called the suspension of V).  Which of
these views should be "notationally invariant"?  Should colimits in Set
be oplimits in the supension of Set? 

Best regards,

-- J"urgen

P.S.  Why are certain categorical notions preferred over their dual
counterparts?  E.g., hardly anyone talks about the Yoneda embedding
of A into [A,Set]^op.

-- 
J"urgen Koslowski       % If I don't see you no more in this world
ITI                     % I meet you in the next world
TU Braunschweig         % and don't be late!
koslowj@iti.cs.tu-bs.de %              Jimi Hendrix (Voodoo Child)

re: co-

[Ross Street]

While our insecurities about "co-" are being aired, I thought I should
admit to even more worries in the case of 2-categories (or bicategories)!
In these terminological matters, I have given up on linguistic correctness
and have also almost given up worrying about mathematical consistency.

Here is the difficulty. Motivation for 2-category theory comes from (at
least) two different directions which often lead to the same basic concepts
yet with different suggestions for terminology for the three other dual
concepts. Each concept has a co-, op-, and coop-version but the good choice
of op or co is not clear at all.

First motivation: We can take the view that our 2-category is foremost a
category with the 2-cells as extra structure (like homotopies in Top).
Then, for example, as pointed out by John Gray in the La Jolla 1965 volume,
Grothendieck was wrong in using "cofibration" for the *2-cell*-reversing
dual of fibration. Compare the situation in  Top  where cofibrations are
the *arrow*-reversing dual of fibrations.  So this leads to "opfibration"
for the *2-cell*-reversing dual of "fibration" (this is unnecessary in  Top
since homotopies are invertible).  However, Grothendieck's terminology has
stuck in some literature.  Using this first motivation, we define products
and coproducts of objects in a 2-category as we would in a category plus an
extra 2-cell condition.

Second motivation: We think of our 2-category  K  as a place to develop
category theory so that arrows f : U --> A  into an object  A  of  K  are
thought of as generalised objects of  A,  and 2-cells into  A  are
generalised arrows of  A.  Take a notion such as monad on  A.  From this
motivation, reversing *2-cells* in  K,  we should get the notion of
"comonad".  This terminology is in conflict with the doctrine developed on
the basis of the first motivation. Of course, "monad" is invariant under
*arrow*-reversal, but there are other concepts which are not.

--Ross

co-

[Paul Taylor]

I seem to have started an avalanche.

There seem to be two contradictory theories - maybe someone 
could make the codictory.

1. co- versus nothing, from trigonometry
2. co- versus contra-

I knew the words covariant and contravariant both before I knew anything
about categories, and from a book which predated (sorry, pre-dated)
category theory, namely
	A.S. Eddington's "Mathematical Theory of Relativity" (1925),
which was the only serious maths book I could find in the town library
in High Wycombe (half way between London and Oxford) when I was at
school.  What it was doing there, I can't imagine.

As to "cofinal", I worked out its etymology for myself, but Saunders
Mac Lane (if he was in fact the culprit) could have re-invented its
etymology instead of generating the confusion.  Besides, (co)final
functors are those between diagram-shapes which give rise to the same
colimits, so the prefix seems reasonable to me.

Paul

Re: co-

[Dr. P.T. Johnstone]

P.S. -- I don't agree with John that "all categorical 'co-'s" 
are of the same kind as "cosine" or "colatitude" in referring to
something complementary (although I suppose that "cohomology"
might be). I can't see any sense in which the opposite of a
category can be regarded as complementary to it.

Peter Johnstone

Re: co-

[Michael Barr]

Well, after reading the various replies, I concede that I was very likely
wrong about cohomolohy (which is generally accepted as the first co-)
being short for contra-homology.  Therefore, complementary homology seems
the best.  Then people started thinking of it as covariant homology
anyway.  Then the idea took hold that co-meant opposite.  Strange, since
now it has essentially reversed meaning.  That's how the language
developes.  It is through such twists and turns that the same
Indo-european root come to mean black in English and white in French (and
other romance languages).  

A few other comments.  The x- and y-coordinates are (or used to be) called
ordinate and abscissa (or vice versa).  For what it's worth.  I know what
efficient means, but I have no idea what it has to do with coefficient.
It stands to reason that Birkhoff & Mac Lane is the opposite of Mac Lane &
Birkhoff.  I hadn't noticed that, but I sure had noticed that I had no
simple way of recalling which were left and which were right cosets.  And
would products be left limits or right limits and why?

Michael

On Sat, 4 Jul 1998, Dr. P.T. Johnstone wrote:

> P.S. -- I don't agree with John that "all categorical 'co-'s" 
> are of the same kind as "cosine" or "colatitude" in referring to
> something complementary (although I suppose that "cohomology"
> might be). I can't see any sense in which the opposite of a
> category can be regarded as complementary to it.
> 
> Peter Johnstone
> 

Re: co-

[Dr. P.T. Johnstone]

Of course the "co-" in "cofinal" is the Latin "cum", as it normally
is in English (if I refer to someone as my co-conspirator, I mean
he is conspiring with me, not against me!). But category-theorists
have got so firmly into the habit of using "co-" as an abbreviation
for "contra-" (except in the terms covariant and contravariant --
I assume they survived because they were widely used before categories
came along) that the "co-" in "cofinal" had to go. As for who killed
it off, the evidence points to Saunders Mac Lane as the guilty party
(see p. 213 of Categories for the Working Mathematician).

Category-theorists at least have the defence that the algebraic
topologists had started using "cohomology" for what should have been
"contrahomology" before categories came along. As Mike Barr mentioned,
Hilton and Wylie tried to encourage the use of "contrahomology" in
their book (1960), but it was probably far too late by then.

I'm surprised that no-one has yet mentioned Barry Mitchell's attempt,
in his book, to "eliminate the words left and right" from the language
of category theory. He did have a scheme for deciding which of a dual
pair of concepts should have the "co-"; unfortunately it led him to use
"adjoint" and "coadjoint" in the opposite sense to that in which most
people had been using them, and so much confusion resulted that everyone
went back to "left adjoint" and "right adjoint".

If it were possible to start afresh with the terminology of category
theory (of course it isn't, as Mike pointed out), I'd be in favour of
using "left" and "right" as much as possible, and eliminating the "co-"s.
(But even this is not guaranteed free from ambiguity. Has anyone apart
from me (and, I suppose, the authors) noticed that the usage of the
terms "left coset" and "right coset" in Mac Lane & Birkhoff's Algebra
is the opposite of that in Birkhoff & Mac Lane?)

Peter Johnstone

co-

[Paul Taylor]

What are the origins of the   co-   prefix, as in coproduct, coequaliser ...,
and who established their use?

Has anybody ever thought through and written down any guidelines on
which of a pair of dual concepts is co-?

Who is reponsible for dropping this prefix from cofinal?
(A mistake, IMHO).

Paul

Re: co-

[James Stasheff]

Surely it goes back at least to cohomology
or further to covariant and contravariant
with their contravariant meanings

************************************************************
	Until August 10, 1998, I am on leave from UNC 
		and am at the University of Pennsylvania

	 Jim Stasheff		jds@math.upenn.edu

	146 Woodland Dr
        Lansdale PA 19446       (215)822-6707	



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


On Fri, 3 Jul 1998, Paul Taylor wrote:

> What are the origins of the   co-   prefix, as in coproduct, coequaliser ...,
> and who established their use?
> 
> Has anybody ever thought through and written down any guidelines on
> which of a pair of dual concepts is co-?
> 
> Who is reponsible for dropping this prefix from cofinal?
> (A mistake, IMHO).
> 
> Paul
> 

Re: co-

[Graham White]

>Surely it goes back at least to cohomology
>or further to covariant and contravariant
>with their contravariant meanings
>
>************************************************************
>	Until August 10, 1998, I am on leave from UNC
>		and am at the University of Pennsylvania
>
>	 Jim Stasheff		jds@math.upenn.edu
>
>	146 Woodland Dr
>        Lansdale PA 19446       (215)822-6707
>
>
>
>	Jim Stasheff		jds@math.unc.edu
>	Math-UNC		(919)-962-9607
>	Chapel Hill NC		FAX:(919)-962-2568
>	27599-3250
>
>
>On Fri, 3 Jul 1998, Paul Taylor wrote:
>
>> What are the origins of the   co-   prefix, as in coproduct, coequaliser
>>...,
>> and who established their use?
>>
>> Has anybody ever thought through and written down any guidelines on
>> which of a pair of dual concepts is co-?
>>
>> Who is reponsible for dropping this prefix from cofinal?
>> (A mistake, IMHO).
>>
>> Paul
>>

I would have thought that `co' in `cofinal' means `together with',
and didn't originally mean `opposite'. There are instances of
this meaning of `co' in, for example,  `coroutine'. And,
of course, `covariant', which is well established in 19th century
invariant theory, where it contrasts with `invariant'
(but I can't remember offhand a 19th cent. use of
`contravariant'). It would be very interesting to see a history
of this terminology.

Graham

Re: co-

[Michael Barr]

Well, I am speculating here.  But FWIW, here goes.  Back in prehistory,
there were covariant and contravariant tensors.  Later on, came homology,
a word with impeccable credentials.  The dual was called cohomology, the
co- doubltess a shortening of contra-.  Very bad choice.  But that's the
way it came.  Peter Hilton pointed out that "homology" should be generic
with cohomology as the covariant version and contrahomology as the
contravariant one.  I think he wrote a book using "homology" and
"contrahomology", a kind of intermediate step.  Good idea, but hopeless,
really.  It reminds me of my pet peeve, which is the use of the horseshoe
for included-in-or-equal.  Thus destroying the analogy with <, as well as
requiring the idiotic horseshoe-plus-not-equal, which does not even appear
in the standard fonts.  So I never use the plain horseshoe for anything.
If everybody did that, then after one generation mathematicians could
start using the horseshoe for proper inclusion.  It will never happen.

As for which is which, that is a harder question.  If D is a diagram,
cone(-,D) is contravariant, but a representing object is called a limit of
D. But limit is covariant in D.  The opposite is true of cocones and
colimits.  Which one is right?  Hard to say?  I call a reflective
subcategory one whose inclusion has a left adjoint, but that has been
called coreflective (although probably not in recent years).  The co- in
cofinal has nothing to do (except perhaps very indirectly) with the one in
colimit.  I think it is like the co- in coordinate.  As such, I see
nothing wrong with final.  Or rather, I don't see that cofinal is any
improvement.  A family of objects in a category is weakly final (or weakly
terminal) if every object in the category has at least one arrow to at
least one object of said family.  Replace both "at least"s by "exactly"
and you have a final (or terminal) family and require the family to be
singleton and you have a final (or terminal) object.  So final ought to be
weakly final and similarly for cofinal, but I don't expect anyone's usage
will change.

Re: co-

[James Stasheff]

OK what is the origin/meaning of co
in
coordinate

perhaps it's time to treat this LESS seriously
as int hold canard
cobras are bras with the eros reversed

************************************************************
	Until August 10, 1998, I am on leave from UNC 
		and am at the University of Pennsylvania

	 Jim Stasheff		jds@math.upenn.edu

	146 Woodland Dr
        Lansdale PA 19446       (215)822-6707	



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

Re: co-

[John R Isbell]

On Fri, 3 Jul 1998, Paul Taylor wrote:

> What are the origins of the   co-   prefix, as in coproduct, coequaliser ...,
> and who established their use?
> 
> Has anybody ever thought through and written down any guidelines on
> which of a pair of dual concepts is co-?
> 
> Who is reponsible for dropping this prefix from cofinal?
> (A mistake, IMHO).
> 
> Paul
> 
      Fragments: (1) Origin, I don't know, but surely cohomology
  is where it started. The term was used very early, 1937 I think,
  by Norman Steenrod in a paper mainly on universal coefficient
  theorems.
                  (2) The idea of putting forward some such
  guidelines was seriously discussed at La Jolla 1965, and I 
  should say that Sammy Eilenberg killed it single-handed. His
  main point was that anything we Americans might propose would
  be absolutely unacceptable in Paris. Verdier was the only
  Frenchman present; he was well thought of but very young.
                  (1 bis) Of course not covariant-contravariant.
                  (3) I'm not sure what "A mistake IMHO" means.
  Of course, the "co" in cofinal is genetically "con" of
  congress, concatenation. I don't have nice illustrations of
  antecedents of co-homology but it is not 'together' like in
  congress & concatenation. But it is dropped in categorical
  contexts because it is a distracting "co".
      John Isbell

Re: co-

[James Stasheff]

I do not understand

(1 bis) Of course not covariant-contravariant.

Surely that is what Steenrod had in mind (subconsciously)?
Remember that covariant-contravariant for diff forms
wass originally referring to change of coordiates rather than maps.

************************************************************
	Until August 10, 1998, I am on leave from UNC 
		and am at the University of Pennsylvania

	 Jim Stasheff		jds@math.upenn.edu

	146 Woodland Dr
        Lansdale PA 19446       (215)822-6707	



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


On Fri, 3 Jul 1998, John R Isbell wrote:

> 
> On Fri, 3 Jul 1998, Paul Taylor wrote:
> 
> > What are the origins of the   co-   prefix, as in coproduct, coequaliser ...,
> > and who established their use?
> > 
> > Has anybody ever thought through and written down any guidelines on
> > which of a pair of dual concepts is co-?
> > 
> > Who is reponsible for dropping this prefix from cofinal?
> > (A mistake, IMHO).
> > 
> > Paul
> > 
>       Fragments: (1) Origin, I don't know, but surely cohomology
>   is where it started. The term was used very early, 1937 I think,
>   by Norman Steenrod in a paper mainly on universal coefficient
>   theorems.
>                   (2) The idea of putting forward some such
>   guidelines was seriously discussed at La Jolla 1965, and I 
>   should say that Sammy Eilenberg killed it single-handed. His
>   main point was that anything we Americans might propose would
>   be absolutely unacceptable in Paris. Verdier was the only
>   Frenchman present; he was well thought of but very young.
>                   (1 bis) Of course not covariant-contravariant.
>                   (3) I'm not sure what "A mistake IMHO" means.
>   Of course, the "co" in cofinal is genetically "con" of
>   congress, concatenation. I don't have nice illustrations of
>   antecedents of co-homology but it is not 'together' like in
>   congress & concatenation. But it is dropped in categorical
>   contexts because it is a distracting "co".
>       John Isbell
> 
> 

Re: co-

[Peter Selinger]

I would guess that the oldest use of co- in mathematics is to mean
"complement of an angle", as in cosine, cotangent, etc.  Encyclopedia
Britannica dates these to 1635. This would be an early justification
of using co- in the sense of "opposite".

The word "complement" itself comes from Latin "complere": to fill up.

The use of co- in the sense of "together, joint" is much more
widespread in everyday language, in words such as coauthor and
coconspirator (notice how the last example is curiously redundant).
This is also the origin of words such as coordinate (1641),
coefficient (ca. 1715), and collinearity (1863).

Best wishes, -- Peter

(Source: Encyclopedia Britannica)


> From Paul Taylor:
> What are the origins of the   co-   prefix, as in coproduct, coequaliser ...,
> and who established their use?
> 
> Has anybody ever thought through and written down any guidelines on
> which of a pair of dual concepts is co-?
> 
> Who is reponsible for dropping this prefix from cofinal?
> (A mistake, IMHO).
> 
> Paul

Re: co-

[James Stasheff]

>This is also the origin of words such as coordinate (1641),
coefficient (ca. 1715), and collinearity (1863).

co-linear I see as together or joint
but what is `ordinate' and what is `effcient'??

************************************************************
	Until August 10, 1998, I am on leave from UNC 
		and am at the University of Pennsylvania

	 Jim Stasheff		jds@math.upenn.edu

	146 Woodland Dr
        Lansdale PA 19446       (215)822-6707	



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

Re: co-

[Peter Selinger]

> From James Stasheff:
> >This is also the origin of words such as coordinate (1641),
> coefficient (ca. 1715), and collinearity (1863).
> 
> co-linear I see as together or joint
> but what is `ordinate' and what is `effcient'??

I am certainly no linguist, but it seems obvious to me that in all
three cases, the prefix was attached before the word entered the
English language, and possibly even before the word acquired its
mathematical meaning. 

Coordinate: from Latin ordinare: to arrange, to put in order.
Coordinates are for "arranging" points in the plane, and they
do this jointly. Compare: coordination.

Coefficient: from Latin efficere: to affect, to produce an effect (?).
Coefficients are parameters that affect some quantity, and usually
there is more than one, so again, they do it jointly.

Does anyone know the actual origin of the word "covariant"? My guess
is that in the original context of tensors on manifolds, it is a
contraction of "coordinate invariant", that is, invariant under
transformations of coordinate systems. If this is true, then it fits
neither the "jointly" nor the "complement" nor the "dual" schemes.
Despite the fact that it came first historically, it would seem that
the word "covariant" is an exception to the otherwise (more or less)
consistent use of the prefix co- in category theory. If one were to
change terminology to eliminate this oddity, it would make little
sense to change the rule to accomodate the exception - rather, one
should rename "covariant" to something more logical like "provariant".

I doubt that it would be worth the effort -- especially since the word
"covariant" only ever seems to appear in parentheses.

Best, -- Peter

Re: co-

[John Duskin]

I seem to remember  the "ordinate" (=y) and "abscissa" (=x) as making
up the cartesian "co-ordinate<italic>s </italic>" of the point (x,y).
"co-efficient" probably comes from terminology for polynomials, with
the "co-" coming from the fact that it was always atttached to a power
of x. And while we are on this we shouldn't forget "direct" and
"inverse" and "inductive" and "projective" limits! It took category
theory (and Mac Lane) to make sense of all of this. 

Re: co-

[John R Isbell]

   Well, the cosine makes a really beautiful story.

   Negative-dimensional chains are not in Lefschetz'
first Colloquium book, but his second. In 1942, so
the co- terminology did not sweep all before it.

   In Lefschetz' first Colloquium book cocycles are

       <pseudocycles>.

   In Steenrod's universal coefficient theorems (1936,
not 1937) cohomology is

       <dual homology>.

   Eilenberg-Steenrod 'Foundations' has a fairly
extensive historical note at the end of Chapter 1.
In particular, they credit 'co' to Whitney, Annals
39 (1938) 397-430 or so (397 is exact). Whitney has
a very brief history on p. 398, tracing the concept
to Alexander 1922, and mentioning a covariant tensor
in Alexander 1935. He says nothing of why he likes
co-.
      John