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
