Subject: categories: co-

From: Paul Taylor <[email-masked]>
Date: Fri, 3 Jul 1998 12:39:19 +0100 (BST)

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

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From: James Stasheff <[email-masked]>
Date: Fri, 3 Jul 1998 13:09:52 -0400 (EDT)

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

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	 Jim Stasheff		[email-masked]

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From: John R Isbell <[email-masked]>
Date: Fri, 3 Jul 1998 15:37:48 -0400 (EDT)

> Who is reponsible for dropping this prefix from cofinal?
      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

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From: Graham White <[email-masked]>
Date: Fri, 3 Jul 1998 20:40:02 +0100

>with their contravariant meanings

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

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From: Michael Barr <[email-masked]>
Date: Fri, 3 Jul 1998 15:28:57 -0400 (EDT)

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.

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From: Peter Selinger <[email-masked]>
Date: Sat, 4 Jul 1998 17:02:15 +0200 (MET DST)

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:

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From: James Stasheff <[email-masked]>
Date: Sat, 4 Jul 1998 10:07:02 -0400 (EDT)

I do not understand

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.

	146 Woodland Dr

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From: James Stasheff <[email-masked]>
Date: Sat, 4 Jul 1998 10:09:29 -0400 (EDT)

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

	146 Woodland Dr

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From: John R Isbell <[email-masked]>
Date: Sat, 4 Jul 1998 13:33:23 -0400 (EDT)

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

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From: James Stasheff <[email-masked]>
Date: Sun, 5 Jul 1998 07:52:45 -0400 (EDT)

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

	146 Woodland Dr

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From: John R Isbell <[email-masked]>
Date: Sun, 5 Jul 1998 13:12:25 -0400 (EDT)

I can't fit this

> but what is `ordinate' and what is `effcient'??

                       remark to coefficients, but when I
studied analytics in 1945, x was the abscissa and y the
ordinate. Presumably z would be the coordinate.
    John

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From: Peter Selinger <[email-masked]>
Date: Sun, 5 Jul 1998 20:10:59 +0200 (MET DST)

> From James Stasheff:

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

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From: John Duskin <[email-masked]>
Date: Sun, 5 Jul 1998 17:24:31 -0400

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.