Subject: categories: Terminology again

From: jean benabou <[email-masked]>
Date: Fri, 30 Dec 2005 17:29:15 +0100

In a previous mail I said that I was strongly opposed to the 
replacement of "cartesian" and "co-cartesian" maps by "prone" and 
"supine" ones for linguistic, mathematical and ethical reasons which I 
was ready to explain in detail if I was asked to do so.
I have waited a few days to see what reactions I  would get. So far, my 
position has been supported by Peter May and Eduardo Dubuc on ethical 
and/or linguistic grounds, and Keith Harbaugh has asked me what 
"ethical"  issue is involved, and more generally to clarify my position 
on the mailing list.

Although the ethical issues are for me the most important, I shall 
postpone them to another mail, if the moderator of this list permits me 
to do so, and I shall concentrate to-day on the mathematical an 
linguistic reasons of my opposition.
I am no linguist, but I am a mathematician, so THE MATHEMATICS WILL 
COME FIRST, and I would like to make this  text more "palatable" and 
less "negative" by introducing some genuinely new ideas, which some of 
you might find of interest, and which are relevant in this debate.

§1- DEFINITIONS

Let  P: C ---> B  be a a functor.
1.1-  A map v of C is VERTICAL (forP)   if  P(v) is an identity. I 
denote by  V(P) or simply V the subcategory of  C which has the same 
objects as C and as maps the vertical ones.
Every identity is vertical, and if in a commutative triangle in C two 
of the maps are vertical so is the third.

1.2- Let f be a map f of C. We shall say that f  is :
(i)  CARTESIAN if for every pair (g,b) with g in C , b in B,  and 
P(g)=P(f).b there exists a unique map h in C such that:   P(h)=b  and  
g=f.h.
(ii)  PRECARTESIAN If the previous condition is satisfied only when b 
is an identity,
(iii)  HORIZONTAL if every vertical map is orthogonal to f . (that 
would probably be "prone" in the proposed new terminology)
I shall denote by  K(P), PreK(P) and H(P) the classes of maps of C 
which are cartesian, precartesian and horizontal, and abbreviate by K, 
PreK,  and H  if  P  is fixed.

1.3 - The functor P is a fibration (resp. a PREFIBRATION)  if for every 
pair (b,X) where b is a map of B, and X an object of C and 
P(X)=Codom(b)  there exists a cartesian (resp.precartesian) map f: Y 
--->X  such that  P(f)=b

1.4 - If X is an object of C and  b: J--->P(X)  we denote Pl(b,X) the 
category of  P-LIFTINGS of b with codomain X with objects the maps f: 
Y--->X  of C such that  P(f)=b, a morphism from f to f': Y'--->X  is a 
vertical map  v: Y--->Y'  such that  f.v=f' .
I shall say that P is a HOMOTOPY PREFIBRATION if all the categories  
Pl(b,X) are connected (which of course imply non empty), the motivation 
of the name is that any two liftings are "homotopic".

1.5 - Let  P: C --->  B  and  P': C' ---> B  be two functors and  F; C 
---> C'  be a functor over B i.e. such that P'.F=P. Such an F obviously 
preserves and reflects vertical maps for P and P'. Moreover if f: Y 
--->X is in Pl(b,X) , F(f): F(Y) ---> F(X) is in P'l(b,F(X))  hence F 
induces functors
           Fl(b,X): Pl(b,X) --->P'l(b,F(X)) , f  l--->F(f)  , for all 
"compatible" pairs (b,X)
I shall say that F is a CARTESIAN FUNCTOR if all the functors Fl(b,X) 
are final. (No assumption is made on  P  or P').

§2-REMARKS "EN VRAC" ABOUT THESE DEFINITIONS (or, let's do a little bit 
of mathematics!)

2.1- Let P: C --->B be a functor  (we assume NOTHING on P) Then:
(i) Every cartesian map is horizontal i.e. K(P) is contained in H(P).
The converse need not be true, for example if all vertical maps are 
iso's i.e. all the fibers of P are groupoïds, then all maps of C are 
horizontal, and "co-horizontal" And one can construct a P where C and B 
are finite posets where no map, except of course the identities, is 
pre-cartesian or pre-cocartesian, let alone cartesian or cocartesian
(ii) THEOREM. If P is a homotopy- prefibration, then cartesian 
coincides with horizontal, i.e.  H(P)=K(p).
(It is not completely trivial)

The definitions 1.4 and 1.5 are genuinely new, and might seem 
surprising, the following remarks will give a very small idea of what 
can be done with them

2.2- A cartesian functor preserves precartesian maps :
Because f: Y --->X is in  PreK(P)  iff it is a final object of  
Pl(P(f),X) , and final functors preserve final objects

However if P and P' are are arbitrary functors such a preservation is 
not enough to insure that F is cartesian because there might not be 
"enough" precartesian maps in C. But cartesian functors have so far 
NEVER been used except between prefibrations, and in that  case our 
definition coincides with the usual one because we have:

2.3- If P and P' are prefibrations,  F  is cartesian iff  it preserves 
precartesian maps.(It suffices in fact that P is a prefibration)

I like to make the following "analogy" : if  S and T are topological 
spaces a continuous function  f: S --->T preserves convergent 
sequences, if X is metrisable, this is enough to insure the continuity 
of f

2.4 - Cartesian functors are closed under composition, and every 
equivalence over B is cartesian.
In fact we have much better, namely.

2.5 - If a functor F over B has a left adjoint then F is cartesian

2.6 - Homotopy-prefibrations are stable by composition.
This seems "harmless" and trivial, but it is neither. It is well known 
that fibrations are stable by composition, but it is probably a little 
less well known, because I have never seen a statement to that effect, 
that prefibrations  ARE NOT .

2.7 - Homotopy-prefibrations  (h-p) are special cases of cartesian 
functors, because P. C --->B  is a h-p iiff it is a cartesian functor: 
(C,P) --->(B,IdB).
(This of course is no longer true if h-p is replaced by prefibration or 
fibration)
 From this it follows that if  F:(C,P) --->(C',P') is cartesian and  P' 
is a h-p so is  P.

2.8 - An important feature of h-p is that pointwise Kan extensions 
along such P's can be computed fiberwise. Moreover this property 
characterizes h-p' s.
In particular such a P is final iff all its fibers are connected, and 
it is flat iff  it's fibers are cofiltered.
The previous results are special cases of properties true for arbitrary 
cartesian functors.

2.9 - REMARK : Homotopy prefibrations are but ONE example of  
MEANINGFUL generalizations of fibrations. I have considered many 
others, all with important mathematical examples, here are some: (for a 
functor  P: C --->B)
(i) The categories Pl(b,X) are filtered
(ii) Each connected component of such a category has a final object
(iii)  Each connected component is filtered
In (ii) and (ii)  P is not even a homotopy prefibration, but in all 
these cases the general definition of cartesian functor given in 1.5  
is the "correct" one and gives the expected results.

§3 LINGUISTICO-MATHEMATICAL REMARKS

3.1 - OK, let us try "prone" for "cartesian", what about the 
precartesian maps, "preprone" ? They have nothing to do with a 
weakening of orthogonality to V(P), which we shall examine in 3.3. What 
about cartesian functors, "prone functors"? What about maps which are 
both cartesian and cocartesian, such that e.g. the iso's, prone and 
supine? A very uncomfortable position you'll grant me. I am no acrobat, 
I tried it, I hurt my back and stomach, had to stand up, and ended 
up...vertical!

3.2 - The proposed terminology is based ON A BIG MATHEMATICAL MISTAKE, 
namely: confusing cartesian and horizontal, which in general do NOT 
coincide, as shown in 2.1. Unless of course there no other functors but 
fibrations, or if there are, the terminology should not be compatible 
with them. Well I, and probably other persons, think that there are, 
know that there are, and that they deserve to be studied, were it only 
to have a better understanding of fibrations. In 2.9 I gave a few 
examples of such functors. If there were ONLY fibrations,  how would 
one express the fact that a prefibration where all the fibers are 
groupoids is a fibration?

3.3 - Even for fibrations there are interesting maps which are neither 
vertical nor cartesian and that one might want to study. Let me give an 
example. Both cartesianness and horizontality assume the existence and 
uniqueness of maps satisfying certain conditions. What about those 
where we drop existence and keep uniqueness.
Following Peter Freyd's suggestion, let me call them 
quasi-cartesian(QK), and quasi-horizontal(QH),and see what they are. A 
map f: Y --->X  is  QK  (rep. QH) iff  for every  parallel pair  
(g,g'): Z===>Y  coequalized by f, if P(g)=P(g') (resp.if g.v=g'.v for 
some vertical map v) then g=g'
Even in the case of fibrations, where K=H, QK is only contained in QH 
but not equal.This can be seen in the most trivial case, where B=1, and 
all maps are vertical. A map f is QK iff it is a mono, it is QH iff  
for every pair of maps (g,g')  WHICH CAN BE EQUALIZED,  fg=fg' implies  
g=g'.
Now if cartesian=prone, QK will have to be "quasi-prone", a strange 
position again, but never mind. However, how should we call QH ?

3.4- I can speak, read, and write a little bit of English, but I am 
French and might someday have the preposterous idea to lecture on 
fibered categories in French.  Of course only in France, and to an 
audience uniquely composed of french persons. Perhaps MM Taylor and 
Johnstone, could suggest adequate french translations for prone and 
supine, which I can't  seem to find. And they should be ready to do the 
same thing for German, Italian, Spanish, and many other languages.

No such problems with cartesian of course, because cartesian.... is 
cartesian  is cartesian is cartesian!

3.5- By now many thousands of pages have been written in various 
languages using "cartesian", and many hundreds are being prepared, or 
ready to be published, using the same word. What should be done with 
all that past or future rubbish, now that we have received THE LIGHT 
and the WORD(S)?

§4 TEMPORARY CONCLUSION

I apologize for such a long mail, but I wanted also to show, among 
other things , that it is possible to handle new and relevant 
mathematical notions by introducing a SINGLE new word, namely; 
"homotopy prefibration" , which has a  clear intuitive content, and 
moreover is easy to translate in most languages.

I have given many arguments to explain my position, and I have many 
more. But for the moment, I'd like to know the arguments of the persons 
in favor of these changes, PRINCIPALLY, of course, those of Paul Taylor 
and Peter Johnstone. If it is only the "joke" aspect, I  want to add 
that I do also like jokes, very much, perhaps not the same as theirs.. 
I even used to compete with Sammy, who was an expert, about who'd know 
some jokes the other didn't.

When this mail was almost completely finished, I found the reaction of 
Vaughan Pratt from which I quote:

  "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?"

I want to be clear on that matter. I have no objection to "frivolous" 
naming of NEW concepts by the person or persons who DISCOVERED or 
INVENTED them. But I object VERY STRONGLY to "renaming" well 
established concepts, used for more than 40 years by the mathematical 
community, even if the new names were NOT frivolous, and especially if 
such a renaming is made by persons who have made no MAJOR
contribution to the development of the field of FIBERED CATEGORIES.

As a side remark, I have no problem whatsoever to translate in French : 
"strange", "charm", "beauty", "quark", "sober" or "bottom". And to be 
"frivolous", even if it's not so easy in a foreign language, 
"homotopy's bottom" came ages before Scott's, and "Galois connection" 
ages before the "french" one.

Since my english is not too good, in particular I knew only "the other" 
meaning of "supine",  I'll borrow, a bit freely, from "a good author" I 
admire a lot, and remind that:

                Men gave names to many animals
                In the beginning, in the beginning
                Men gave names to many animals
                In the beginning, long time ago.

Best wishes to all, Jean

---

From: David Yetter <[email-masked]>
Date: Sat, 31 Dec 2005 22:01:54 -0600

[Note from moderator: This is to let you know that I am invoking the
(not recently used) 48 hour rule for this subject: postings received
by noon on Wednesday will be sent; after that the discussion may
obviously continue, but not on the list.
Best wishes to all for 2006, Bob Rosebrugh]

> contribution to the development of the field of FIBERED CATEGORIES.

I will second Jean's remarks excerpted above, with a sole exception:  I
have no objection to
the renaming of a well-established concept in honor of the person(s)
who discovered or invented them (or, if a second name is attached, who
first made its importance clear).

Best wishes to all for the new year,
David Yetter