Date: Fri, 3 Jul 1998 12:39:19 +0100 (BST)
From: Paul Taylor
Subject: categories: co-
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
Date: Fri, 3 Jul 1998 13:09:52 -0400 (EDT)
From: James Stasheff
Subject: categories: Re: co-
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
>
Date: Fri, 3 Jul 1998 15:28:57 -0400 (EDT)
From: Michael Barr
Subject: Re: categories: co-
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.
Date: Fri, 3 Jul 1998 20:40:02 +0100
From: Graham White
Subject: categories: Re: co-
>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
Date: Fri, 3 Jul 1998 15:37:48 -0400 (EDT)
From: John R Isbell
Subject: categories: Re: co-
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
Date: Fri, 3 Jul 1998 15:44:49 -0400 (EDT)
From: John R Isbell
Subject: categories: re: co-: P.S.
In the last sentence of my Re: categories: co-,
insert 'from "cofinal"'.
Date: Sat, 4 Jul 1998 10:09:29 -0400 (EDT)
From: James Stasheff
Subject: categories: Re: co-
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
Date: Sat, 4 Jul 1998 10:07:02 -0400 (EDT)
From: James Stasheff
Subject: categories: Re: co-
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
>
>
Date: Sat, 4 Jul 1998 11:36:35 -0400 (EDT)
From: John R Isbell
Subject: categories: Re: co-
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
From: Peter Selinger
Subject: categories: Re: co-
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:
> 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
Date: Sat, 4 Jul 1998 13:33:23 -0400 (EDT)
From: John R Isbell
Subject: categories: Re: co-
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
.
In Steenrod's universal coefficient theorems (1936,
not 1937) cohomology is
.
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
Subject: categories: Re: co-
Date: Sat, 4 Jul 1998 18:30:40 +0100 (BST)
From: "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
Subject: categories: Re: co-
Date: Sat, 4 Jul 1998 18:40:15 +0100 (BST)
From: "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
Date: Sun, 5 Jul 1998 07:52:45 -0400 (EDT)
From: James Stasheff
Subject: categories: Re: co-
>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
Date: Mon, 6 Jul 1998 16:07:47 +0100 (BST)
From: Ronnie Brown
Subject: categories: co etc
Re Hilton and Wylie: it really was a landmark in its time, quite readable.
But the co- contra- change, together with contravariant functors written
on the right, became contrafusing.
Ronnie
From: Peter Selinger
Subject: categories: Re: co-
Date: Sun, 5 Jul 1998 20:10:59 +0200 (MET DST)
> 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
Date: Sun, 5 Jul 1998 17:24:31 -0400
From: John Duskin
Subject: categories: Re: co-
I seem to remember the "ordinate" (=y) and "abscissa" (=x) as making
up the cartesian "co-ordinates " 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.
Date: Mon, 06 Jul 1998 09:16:48 -0400
From: Charles Wells
Subject: categories: Left and right
>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?)
Not all lefts and rights are the same. Left adjoint refers to the fact
that arrows FROM a value of the left adjoint into an object correspond to
arrows INTO the value of the right adjoint at that object. Since English
is written left to right, Hom(A,B) means arrows from A to B, so in the
equation Hom(FA, B) = Hom(A,UB) the F winds up on the left side of the hom
set. This is a natural name given the way we write our language, and so it
is not hard to reconstruct what the phrases left and right adjoint mean.
On the other hand the left and right in "left inverse" and "right inverse"
depend on the order in which we write composition, and that is independent
of the way we write our language. I for one can never remember which is
which, a learning disability no doubt accounted for by the fact that I
worked on semigroups before I became a category theorist, leaving me
without a default way to write composition. In any case, I would like
names such as retraction and split.
Charles Wells, Department of Mathematics, Case Western Reserve University,
10900 Euclid Ave., Cleveland, OH 44106-7058, USA.
EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893.
FAX: 216 368 5163. HOME PHONE: 440 774 1926.
HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html
Date: Sun, 5 Jul 1998 13:12:25 -0400 (EDT)
From: John R Isbell
Subject: Re: categories: Re: co-
I can't fit this
On Sun, 5 Jul 1998, James Stasheff wrote:
> >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'??
>
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
Date: Mon, 6 Jul 1998 12:02:12 -0400 (EDT)
From: Michael Barr
Subject: categories: Re: co-
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
>
Date: Mon, 6 Jul 1998 19:15:35 +0100 (BST)
From: Paul Taylor
Subject: categories: co-
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
Subject: categories: A sense in which C^op is complementary to C
Date: Mon, 06 Jul 1998 14:50:29 -0700
From: Vaughan Pratt
From: "Dr. P.T. Johnstone"
>I can't see any sense in which the opposite of a
>category can be regarded as complementary to it.
Peter's book "Stone spaces" (CUP 1982) studies many categories that
obey the rule that the signature of the opposite of C is the complement
of the signature of C, with respect to both existence and direction
of sup's and inf's of cardinalities 0, finite nonzero, and infinite.
Namely, if sup's of a given arity exist in C then inf's of that arity
do not exist in C^op, and conversely (at least in the examples).
The extremal pair of opposite categories in this regard are CABA,
complete atomic Boolean algebras, which has all inf's and sup's, dual to
Set having none (but for this purpose we could as well take StoneDLat,
dual to Poset). "In the middle" is CSLat, complete semilattices,
having sup's of all cardinalities and no inf's, dual to itself via a
duality that interchanges sup's and inf's (not to be confused with the
equivalence that performs this interchange).
Top and bottom as zeroary sup and inf can be moved independently from
CABA to Set to give pointed and bipointed sets dual to CABA's lacking
one or both constants 0 and 1.
Likewise binary (and hence ternary etc.) sup and/or inf can be moved from
CABA to Set. Moving all finitary sup's for example turns Set into SLat
(semilattices, specifically of the meet kind) and CABA into StoneSLat,
aka algebraic lattices.
By viewing Stone topology as the manifestation of infinite sup's and
inf's, the duality of Stone (Stone spaces) and Bool (Boolean algebras)
can be understood as the migration of the infinitary sup's and inf's
of CABA to Set, turning CABA into Bool and Set into Stone. From this
viewpoint the ability to present limits in Top follows by duality from
the optional removal in CABA of infinitary inf's to yield Top^op, while
Frm (frames) as Loc^op (locales) tightens this up by removing "optional".
Chu(Set,2) provides a uniform setting for this rule. All the above
categories C have a "common" full concrete embedding in Chu(Set,2)
in which the points remain unchanged (concreteness) while the states
or opens are taken to be the morphisms of C to the realization of the
schizophrenic object 2 as an object of C. In this setting the above
tendency (but not the converse part) can be made a theorem about Chu
spaces over 2 as follows, at least for the finitary operations.
Theorem 0: If a space A has constant 0 then its dual A\perp does not
have constant 1 (and dually with 0 and 1 interchanged).
Proof. A row of all 0's precludes a column of all 1's (the impossibility
of having an irresistible force and an immovable object).
Theorem 2: If A has all binary joins then the only binary meets existing
in A\perp are of comparable elements.
Proof. Suppose x,y are incomparable columns of A (points of A\perp)
whose meet x&y is a column of A. Then there exist rows a,b of A such that
x y x&y
a 0 1 0
b 1 0 0
is part of A's matrix. Now form the join of a and b:
a|b 1 1 0
This contradicts x&y being the meet of x and y. QED
(Apropos of the comparability issue, note that the category of finite
chains with bottom, with morphisms all monotone bottom-preserving
functions, is self-dual.)
I don't know how to state the corresponding theorem if any for infinitary
sup's and inf's. And I have no idea how to generalize all this to
Chu(Set,3) and beyond.
I became aware of this complementarity principle for opposite categories
around 1990, and contemplating it led my student Vineet Gupta and myself
to (ordinary) Chu spaces in 1992, at which point we learned that they
were not new (but not well known).
Vaughan Pratt
Date: Tue, 7 Jul 1998 10:49:43 +1000
From: street@mpce.mq.edu.au (Ross Street)
Subject: categories: re: co-
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
Date: Tue, 7 Jul 1998 08:07:35 -0400 (EDT)
From: James Stasheff
Subject: categories: Re: co etc
inspite of the book being coproductive
************************************************************
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 Mon, 6 Jul 1998, Ronnie Brown wrote:
>
> Re Hilton and Wylie: it really was a landmark in its time, quite readable.
> But the co- contra- change, together with contravariant functors written
> on the right, became contrafusing.
>
> Ronnie
>
>
Date: Tue, 7 Jul 1998 08:09:41 -0400 (EDT)
From: James Stasheff
Subject: categories: Re: Left and right
Not to mention the notation Ext(A,B) in which B is extended BY A
************************************************************
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 Mon, 6 Jul 1998, Charles Wells wrote:
>
> >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?)
>
> Not all lefts and rights are the same. Left adjoint refers to the fact
> that arrows FROM a value of the left adjoint into an object correspond to
> arrows INTO the value of the right adjoint at that object. Since English
> is written left to right, Hom(A,B) means arrows from A to B, so in the
> equation Hom(FA, B) = Hom(A,UB) the F winds up on the left side of the hom
> set. This is a natural name given the way we write our language, and so it
> is not hard to reconstruct what the phrases left and right adjoint mean.
>
> On the other hand the left and right in "left inverse" and "right inverse"
> depend on the order in which we write composition, and that is independent
> of the way we write our language. I for one can never remember which is
> which, a learning disability no doubt accounted for by the fact that I
> worked on semigroups before I became a category theorist, leaving me
> without a default way to write composition. In any case, I would like
> names such as retraction and split.
>
>
> Charles Wells, Department of Mathematics, Case Western Reserve University,
> 10900 Euclid Ave., Cleveland, OH 44106-7058, USA.
> EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893.
> FAX: 216 368 5163. HOME PHONE: 440 774 1926.
> HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html
>
Date: Tue, 07 Jul 1998 15:27:14 -0400
From: Zhaohua Luo
Subject: categories: Re: A sense in which C^op is complementary to C
A category is called " left " if it has a strict initial object and any map
with an initial domain is regular (in the general sense). The opposite of a
left category is called a " right " category. The key fact here is that a left
category is never right unless it is trivial (i.e. any map is an isomorphism).
Since may natural categories are either left or right, this enable us to
introduce the "relative version" of a categorical concept with a dual. The
following short note
On Left and Right Categories
is available at
www.azd.com/lrcat.html
which is inspired by Vaughan Pratt comments:
Vaughan Pratt wrote:
> From: "Dr. P.T. Johnstone"
> >I can't see any sense in which the opposite of a
> >category can be regarded as complementary to it.
>
> Peter's book "Stone spaces" (CUP 1982) studies many categories that
> obey the rule that the signature of the opposite of C is the complement
> of the signature of C, with respect to both existence and direction
> of sup's and inf's of cardinalities 0, finite nonzero, and infinite.
> Namely, if sup's of a given arity exist in C then inf's of that arity
> do not exist in C^op, and conversely (at least in the examples).
>
> The extremal pair of opposite categories in this regard are CABA,
> complete atomic Boolean algebras, which has all inf's and sup's, dual to
> Set having none (but for this purpose we could as well take StoneDLat,
> dual to Poset). "In the middle" is CSLat, complete semilattices,
> having sup's of all cardinalities and no inf's, dual to itself via a
> duality that interchanges sup's and inf's (not to be confused with the
> equivalence that performs this interchange).
>
> Top and bottom as zeroary sup and inf can be moved independently from
> CABA to Set to give pointed and bipointed sets dual to CABA's lacking
> one or both constants 0 and 1.
>
> Likewise binary (and hence ternary etc.) sup and/or inf can be moved from
> CABA to Set. Moving all finitary sup's for example turns Set into SLat
> (semilattices, specifically of the meet kind) and CABA into StoneSLat,
> aka algebraic lattices.
>
> By viewing Stone topology as the manifestation of infinite sup's and
> inf's, the duality of Stone (Stone spaces) and Bool (Boolean algebras)
> can be understood as the migration of the infinitary sup's and inf's
> of CABA to Set, turning CABA into Bool and Set into Stone. From this
> viewpoint the ability to present limits in Top follows by duality from
> the optional removal in CABA of infinitary inf's to yield Top^op, while
> Frm (frames) as Loc^op (locales) tightens this up by removing "optional".
>
> Chu(Set,2) provides a uniform setting for this rule. All the above
> categories C have a "common" full concrete embedding in Chu(Set,2)
> in which the points remain unchanged (concreteness) while the states
> or opens are taken to be the morphisms of C to the realization of the
> schizophrenic object 2 as an object of C. In this setting the above
> tendency (but not the converse part) can be made a theorem about Chu
> spaces over 2 as follows, at least for the finitary operations.
>
> Theorem 0: If a space A has constant 0 then its dual A\perp does not
> have constant 1 (and dually with 0 and 1 interchanged).
>
> Proof. A row of all 0's precludes a column of all 1's (the impossibility
> of having an irresistible force and an immovable object).
>
> Theorem 2: If A has all binary joins then the only binary meets existing
> in A\perp are of comparable elements.
>
> Proof. Suppose x,y are incomparable columns of A (points of A\perp)
> whose meet x&y is a column of A. Then there exist rows a,b of A such that
>
> x y x&y
> a 0 1 0
> b 1 0 0
>
> is part of A's matrix. Now form the join of a and b:
>
> a|b 1 1 0
>
> This contradicts x&y being the meet of x and y. QED
>
> (Apropos of the comparability issue, note that the category of finite
> chains with bottom, with morphisms all monotone bottom-preserving
> functions, is self-dual.)
>
> I don't know how to state the corresponding theorem if any for infinitary
> sup's and inf's. And I have no idea how to generalize all this to
> Chu(Set,3) and beyond.
>
> I became aware of this complementarity principle for opposite categories
> around 1990, and contemplating it led my student Vineet Gupta and myself
> to (ordinary) Chu spaces in 1992, at which point we learned that they
> were not new (but not well known).
>
> Vaughan Pratt
From: Koslowski
Subject: categories: re: co-
Date: Wed, 8 Jul 1998 13:10:05 +0200 (MET DST)
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)
Date: Wed, 08 Jul 1998 14:39:02 -0500 (EST)
From: Fred E J Linton
Subject: categories: RE: co-
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
Subject: categories: RE: co-
Date: Wed, 08 Jul 1998 21:04:34 -0700
From: Vaughan Pratt
From: Fred E J Linton
>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
Subject: categories: Re: A sense in which C^op is complementary to C
Date: Wed, 08 Jul 1998 22:25:50 -0700
From: Vaughan Pratt
From: Zhaohua Luo
>A category is called " left " if it has a strict initial object and any map
>with an initial domain is regular (in the general sense). The opposite of a
>left category is called a " right " category.
This division of categories into left, right, and other caters for only
a very special case of the general phenomenon "signature of opposite is
complement of signature", namely where *all* the algebra is on the right,
call this C\op. The more general situation permits the algebra to be
split between C and C\op, e.g. CSLat which puts the sups (say) in C and
sends the infs off to C\op, which turn into sups as they "pass through
the mirror." Many other partitions of algebra are possible that similarly
do not fit this left-right classification and so fall under "other".
Vaughan Pratt
Date: Wed, 15 Jul 1998 10:50:34 -0300
From: Robert Dawson
Subject: categories: Re: co-
----------
> From: Paul Taylor
> 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
Date: Thu, 10 Sep 1998 20:50:38 +1000 (EST)
From: maxk@maths.usyd.edu.au (Max Kelly)
Subject: categories: the origins and use of "co"
Having been out of email contact since late June, what with travels and
hospitals, I have only now read the correspondence in early July on this
subject. I note that the only "co" in the Macquarie dictionary is that
coming from Latin "cum', of which the "co" in "cosine" is a special case,
in that it is short for "complement".
I don't recall ever knowing the origin of "colimit" and so on, but I can
report an interesting observation by Sammy Eilenberg (in private conversation,
when we were collaborating) on why projective limits are "limits" and
inductive limits are "colimits", and not the reverse. His point was that
L is a limit in the category A iff, for each a in A, the set A(a,L) is a
limit in Set; while C is a colimit in A iff, for each a in A, the set A(C,a)
is a limit in Set - not a colimit. So one needs limits in Set even to DEFINE
colimits in A. If you like, God made limits, while colimits are Menschenwerk.
As for "cofinal", where the "co" was originally of the "cum" type, meaning
that the subsequence was "equally final" with the whole sequence, and which
was "confinal" in German, we of the Sydney school - very likely at the same
time as others - saw it as hopelessly confusing in view of the categorical
use of "co", and deliberately changed it to "final" in our writings, with
"initial" for the dual. Note that the notion of a final functor works
beautifully even for WEIGHTED limits, as in my book on enriched categories.
Max Kelly.