Subj:	A name request
Date:        Tue, 02 Jun 92 21:20:34 ADT
From: dbenson@yoda.eecs.wsu.edu (David B. Benson)

Given a pair of arrows with common codomain, the pullback consists of
an object and a pair of pullback projections with the requisite
universal properties.  Now for the question:

	Regular is to equalizer as X is to pullback projections.

What name for a pair of arrows with a common domain should replace X in
the above line?

Thanks,
David
==========================================================================
Subj:	Re: A name request
From: street@macadam.mpce.mq.edu.au (Ross Street)
Date: Fri, 5 Jun 92 13:00:10 EST

> 
> Date:        Tue, 02 Jun 92 21:20:34 ADT
> From: dbenson@yoda.eecs.wsu.edu (David B. Benson)
> 
> Given a pair of arrows with common codomain, the pullback consists of
> an object and a pair of pullback projections with the requisite
> universal properties.  Now for the question:
> 
> 	Regular is to equalizer as X is to pullback projections.
> 
> What name for a pair of arrows with a common domain should replace X in
> the above line?
> 
> Thanks,
> David
> 

X = "pullback span" (see March 1972 thesis of Jeanne Meisen supervised
by Jim Lambek at McGill)

or, in the additive case,

X = "relation" (see Peter Hilton's LaJolla paper, 1965).

Regards,
Ross
==========================================================================
Subj:	Re: A name request
From:	Richard Wood <rjwood@cs.dal.ca>
Date:	Fri, 5 Jun 1992 13:58:40 -0300

I assume that by "regular" you mean "regular mono". If that is so
then I think that your X is "difunctional jointly monic pair". Aurelio,
Christina and company have recently looked at difunctional relations in
their work on Mal'cev matters but I do not have an explicit reference for 
you.
The X above is an ugly mouthfull but it makes sense in that for the most
"decent" of categories, toposes, every mono is "regular" suggesting a
usualness or normalness about such monos relative to the paradigm, one
to one function_but difunctionality is not usual.   
To be precise, let X<---R--->A be a jointly monic pair in SET then R
is a pullback of some cospan if and only if, viewed as a relation, R
is "transitive" in the following sense:
	      aRx and bRx and bRy   implies   aRy    .
This interpretation is of course possible in much greater generality
than SET.

Best regards, Richard
==========================================================================
Subj:	A rose by any other name, Mal'cev relations
Date: Mon, 8 Jun 92 15:19:39 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)
 
I used the phrase regular--or effective--epimorphic family for the dual
notion of pair of arrows that were a pushout.  Of course, this is a much
wider notion, so regular epi pair would do for that.  Since it is almost
self-defining and also shorter than some of the alternatives, it is the
term I recommend.
 
Richard points out, in effect, that a subset R in A x B is a
pullback in this sense iff R o R\op o R is in R. That is true in any
topos.  It is also true in any abelian category.  Is it true in the
dual of a topos?  In the dual of sets?  Whatever, it is an exactness
condition (that any such subset be a pullback) that might turn out
to be interesting.  Conversely, it could turn out to be equivalent
(probably with some existence of limits and/or colimits) to
effective equivalence relations.  Note that it implies that R o R\op
and R\op o R are transitive (they are obviously symmetric).
 
Michael
++++++++++++++++++++++++
Date: Tue, 9 Jun 92 10:57:05 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

This is a follow  up to my post from yesterday.
Let me say that a subobject R in A x B is a Mal'cev relation if
R o R\op o R is in R.  Call a Mal'cev relation effective if there is a
pullback
                    R ------> A
                    |         |
                    |         |
                    |         |
                    v         v
                    B ------> C
 
Yesterday, I pointed out, in effect, that in toposes and abelian
categories, Mal'cev relations are effective and asked if it was true in
the dual of sets.  The following additional facts have now come to
light:
 
1. Mal'cev relations are not effective in the dual of sets.
(Counter-example at end).
 
2. In a Mal'cev category, every relation is Mal'cev (trivial), but
not always effective (see 1).
 
3. If every Mal'cev relation is effective, then so is every equivalence
relation.  Since equivalence relations are effective in the opposite of
sets, it follows from 1 that the converse is false.  Thus effectiveness
of Mal'cev relations is strictly stronger than that of ERs.
 
4. A PER (on a single object A) is a Mal'cev relation, but the converse
is not true (trivial counter-example in groups, in which, from 2, every
relation is Mal'cev).
 
Thus the condition of effectiveness of Mal'cev relations is an exactness
condition that is strictly stronger than that of ERs.  Since it is true
in any topos, it must have something to do with disjoint universal sums.
It wouldn't surprise me if it had something to do with effective unions.
 
Here is the counter-example.  If there is a pullback as above, then C
can be taken to be the pushout.  Thus if sets\op satisfy the condition,
so does the opposite of finite sets, which is equivalent to finite
boolean algebras.  In that category, the following is a pushout and not
a pullback (there is no choice in the maps since 2 is initial and 1 is
final):
                    2 ------> 4
                    |         |
                    |         |
                    |         |
                    v         v
                    1 ------> 1
 
Since Boolean algebras is a Mal'cev category, all relations are Mal'cev
(it is trivial to see that this one is in any case).
 
It follows that effective Mal'cev pair would be another answer to the
original question, but not a good one, I think.
 
Michael
==========================================================================
Subj:	Re: Mal'cev relations
Date: Tue, 9 Jun 92 21:37:51 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

Please change the line that says that in a Mal'cev category, every relation is
Mal'cev to one that says that if an equational category has a Mal'cev
operation, then every relation is Mal'cev.

Thanks, Mike
==========================================================================
Subj:	Mal'cev relations and dinaturality
From: <mxh@dcs.edinburgh.ac.uk>
Date: 10 Jun 92 13:12 +0100

As a followup to the discussion on Mal'cev relations these days I
thought it might be a good idea to describe some ideas I had recently
concerning these. They seem quite close to Peter Freyd's structors, so
I don't claim originality.

Let BB be a ccc with finite limits. A pullback square
(a:X->A,b:X->B,p:A->Y,q:B->Y) is called a Mal'cev relation between A
and B, (by a slight distortion of Michael Barr's definition). Are
there any good terms for X and Y? One might call them kernel and
image, perhaps.

The Mal'cev relations in BB form a ccc with finite limits (morphisms
being given by hexagons.). Now given a class of -+ bifunctors on BB we
can axiomatise an action on Mal'cev relations as follows: If
R:=(a:X->A,b:X->B,p:A->Y,q:B->Y) is a M. rel between A and B and
F:BB\op x BB -> BB is in the class of bifunctors, then we assign to
this situation a Mal'cev relation F*R between FAA and FBB such that
the following axioms are satisfied:

1) (F x G)*R =~= F*R x G*R
2) (F => G)*R =~= F*R => G*R
3) If F_R is the relation between FAA and FBB defined by the span
(Fpa:FYX->FAA, Fqb:FYX->FBB) and F^R is the relation defined by the
cospan (Fap:FAA->FXY, Fqb: FBB->FXY) then F_R \subset F*R \subset F^R.

In 3) the term \subset and "relations are used in an informal way.
What 3) precisely means is that 

		       FAA               FAA
		     /\   \            /\   \
		     /     \           /     \
		    /      \/         /      \/
		   W       FXY      FYX       Z
		    \      /\         \      /\
		     \     /           \     /
		     \/   /            \/   /
		       FBB               FBB

commute for W the kernel and Z the image of F*R. 

=> and x denote twisted exponential and pointwise product for
bifunctors and exponential and cartesian product for Mal'cev
relations. The clauses 1)-3) can easily be generalised to
-^n+^n-functors.  One can show by induction that the class of
bifunctors definable by constants +(if it exists), x and => admit 
an action on Mal'cev relations, so the definition is consistent.

Now given such a class of functors and an action on Mal'cev relations
we can define a notion of dinaturality as follows:

Def: Let F and G be two -+ bifunctors.
A "very strong dinatural transformation" t is then given by a
family of maps t_A:FAA to GAA for each object A in C, such
that for each Mal'cev relation R between A and B the pair 
(t_A:FAA->GAA, t_B:FBB->GBB) is a morphism from F*R to G*R in the
category of Mal'cev relations. I.e. the "famous hexagon"

                                    t_A
			      FAA -------> GAA
			      /\             \
			      /               \ 
			     /                 \
			    /                  \/
			   W                    Z
			    \                  /\
			     \                 /
			      \               / 
			      \/    t_B      /
			      FBB -------> GBB


for W the kernel of F*R and Z the image of G*R.
(end of definition)

These transformations compose and, in effect, form a ccc (with bifunctors
as objects). We may call a polymorphic function parametric if it comes
from such a transformation.

Remark: By letting R := the graph of some morphism it follows
that every "very strong dinatural transfomation" is dinatural in the
Yoneda-MacLane-Bainbridge-Freyd-Scedrov-Scott sense.

Now we are ready to define the corresponding notion of an "end" as a
universal transformation from some constant functor to a bifunctor and
prove that in the case of FXY == (TY=>X)=>Y (for T covariant) such an end
gives the initial T-algebra for T. Moreover one can show that the initial
T-algebra _is_ an end, so that e.g. in the category of sets an end for
FXY=(Y=>X)=>(X=>Y) exists (namely the NNO). In this sense the category
provides a "bad model for polymorphism" looked for by E.Robinson in
that it distinguishes different notions of parametricity. At least as
far as ML polymorphism is concerned.

As a side effect this provides a characterisation of a strong NNO in a
ccc with finite limits by pure equations.

Martin Hofmann
==========================================================================
Subj:	Mal'cev relations
Date: Thu, 11 Jun 92 13:34:12 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

The observations by Martin Hoffman are interesting.  In the interests of
clarity, I do object to his calling the category of pullback squares
the category of Mal'cev relations.  One is an existential structure,
while the other builds the thing that exists into the structure.  It makes
it  into an entirely different category.  Not less interesting, but
different.  It is (somewhat) like the difference between the categories
of complete lattices and complete semilattices.  

One interesting (and trivial) observation is that if Mal'cev relations
are effective, then a quotient of a subobject of a quotient.  This is
(like everything I've said, but forget to mention) in the presence
of sufficient limits and colimits.  Now if one could show that the
amalgamation property held, one would have that a pushout of a mono
is mono in that case.

Michael
==========================================================================
Subj:	Re: Mal'cev relations
Date: Sun, 14 Jun 92 08:08:20 GMT-0400
From: jds@rademacher.math.upenn.edu

the difference beteen the two categories triggers memories of 
discussions with John Moore about the non-category of H-spaces
the problem is with the morphisms
either an H-map is one that respects the two multiplications up
to some unspecified homotopy
OR
an H-map is a pair consisting of a map say f:X --> Y AND a specific
homotopy h_t:X x X --> Y

either way there are problems
thoughts?
jim
==========================================================================
Subj:	Re: Mal'cev relations
Date: Mon, 15 Jun 92 10:05 GMT
From: MAS013@BANGOR.AC.UK

(This is a reply to Jim's thought on H-spaces, not really to Mike's one 
on Malcev relations.)
There would seem to be a need for a discussion of homotopy everything 
models of the theory of categories.  The objects would have a 
composition that was homotopy associative up to arbitrary degree, would 
have homotopy identities, etc.  Vogt back in 1974 provided an example of 
such a thing in the context of homotopy coherent diagams of spaces.  The 
composition morphism needed to be  chosen, but once chosen was homotopy 
associative to infinite degree.  (Jean-Marc Cordier and I have examined 
this in the context of simplicially enriched categories.)  It seems that 
a possible way out is to consider such an infinitely lax category as a 
simplicial class satisfying a weak Kan condition, that condition giving the 
composition. 
Grothendieck in his pursuit of stacks asked for infinitely lax categories 
or groupoids to model homotopy types, and recent work by Kapranov, 
Voevodsky, Steiner etc., as well as a mass of material originating 
"down-under" from Ross, and friends, suggest that the time is 
approaching when an attack on these problems can be made.  Jean-Marc and 
I have a lot of information on doing homotopy coherence in simplicially 
enriched settings including homotopy coherent forms of the Yoneda lemma, 
the interchange law in this setting, and so on.  Unfortunately it is not 
always obvious how to go beyond locally weakly Kan simplicial 
category, where at least the composition is defined, at least, so as to 
be able to handle weakly Kan simplicial classes.

I would much appreciate peoples reactions to this view.

Cheers,

Tim.

(Sender:Tim Porter: 
e-mail: mas013@uk.ac.bangor.vaxc
School of Maths, 
University of Wales at Bangor,
Dean Street,
Bangor,
Gwynedd, LL57 1UT.
U.K.

==========================================================================
Subj:	H-Spaces
Date: Tue, 16 Jun 92 12:56:09 EDT
From: jds@rademacher.math.upenn.edu

Jim Stasheff replying to tim Porter's comments on my H-space query:

Yes, the analog for H-spaces is fine.  If X is not only an H-space
but a strongly homotopy associative (sha) space, then it has a classifying
space BX and the category of sha H-spaces can be defined so that B
is a functor, but at any finite level ...???
==========================================================================
Subj:	Horst Reichel
Date: Fri, 19 Jun 92 12:54:25 -0400
From: cfw2@po.CWRU.Edu (Charles F. Wells)

Does anyone have a current address, email or pmail, for Horst
Reichel?

Thanks, Charles.
==========================================================================
Subj:	Definition of regular categories
Date: Sun, 21 Jun 92 11:34:13 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)
 
Many definitions of regular category have been given.  The original one
required that every arrow have a kernel pair (it will not do to call it
a kernel since that has another, and related, meaning in abelian
categories), that the kernel pair of every arrow have a coequalizer and
that every coterminous pair of arrows of which one is a regular epi (a
coequalizer of a kernel pair) have a pullback in which the arrow
opposite the regular epi is regular epi.  This definition can be both
strengthened and weakened (and has been, including by me), while
retaining its essential intent.
 
Strengthenings:
 
You can assume the category has more (finite) limits or more
coequalizers.  All the way up to all finite limits and finite colimits.
 
Weakenings:
 
You can define a regular epi as I did last fall, in terms of pairs of
arrows.  You can force it to be stable by requiring that it be a cover
in the finest topology for which the representable functors are sheaves.
You can then require that every arrow factor as a regular epi followed
by a mono.
 
I think that if I were starting from scratch, I would use the latter
definition and then strengthen it as needed.  The full embedding theorem
is certainly true in that context.
 
As for examples, that's harder.  I haven't worked out the details, but I
think that finite ordinals and order-preserving maps should be an
example.  If you stick to non-zero ordinals, then this is the standard
simplicial category.  All the monics and all the epics are split and
hence regular in any definition, but I don't think the epis have kernel
pairs, although coequalizers probably exist.  The point is that there is
no reason to constrict the notion, since you can add whatever you need
when you need it.
 
In a regular category, it is certainly equivalent to say that every epi
is regular and that every monic epi is an isomorphism.  Moreover, it
works with the weakest definition, since you need only take an epi and
factor it as a regular epi followed by a mono.  Then the mono is the
second factor of an epi and hence also epic and so if every monic epi is
an ismorphism, the original is, up to isomorphism, the regular epi part.
The converse is even easier.  I thought the definition of pretopos
implied regular.
 
Michael
==========================================================================
Subj:	WANTED: Topos Theory
Date: Wed, 24 Jun 92 11:30:02
From: Steve Vickers <sjv@doc.imperial.ac.uk>

Can anyone in Britain sell me a copy of Peter Johnstone's "Topos Theory"? 
Name your price.

Steve Vickers.
==========================================================================
Subj:	Re: WANTED: Topos Theory and WANTED Basic Concepts...
Date: Fri, 26 Jun 1992 10:57 ADT
From: RDAWSON@HUSKY1.STMARYS.CA

	This must be the Most Wanted Book In The World... do the publishers
realize the demand? To whom it may concern: if anybody brings out an edition
at any reasonable price, I will buy one. Paperback would be nice but not
essential. I will also recommend that our library purchase a copy,. That's 2.
I also know several other people who want copies...
	Note that I am *not* opening bidding against Steve Vickers; this
is intended for use to get the publishers going. But, seriously, somebody
must have contacts to pass this on!
	-Robert Dawson
++++++++++++++++++++++++++
From: dpym@dcs.ed.ac.uk
Date: Fri, 26 Jun 92 15:00:36 BST

In a vein similar to that of Steve Vickers' note: 

Will anyone in the U.K. sell me a copy of `Basic Concepts of 
Enriched Category Theory', by G.M. Kelly ? 

D.J. Pym
==========================================================================
Subj:	Re: WANTED: Topos Theory  ... 
From: John C. Mitchell <jcm@cs.stanford.edu>
Date: Fri, 26 Jun 92 10:55:21 -0700

I did buy a copy of Topos Theory a couple of years ago direct from the
publisher. It turned out to be an "Academic Press Replica Reprint,"
which means a bound Xerox copy. I think the price was
somewhat high, but I don't remember exactly. Despite the poor print
quality, I think it is better to own a copy of this sort than none at
all. A publisher can certainly do this without any substantial inventory
costs, etc. On the other hand, without going into details, there
could be other ways to obtain a copy of similar quality ...

John Mitchell
==========================================================================
Subj:	the 2- (n-?) category of chain complexes 
From: dyetter@math.ksu.edu (David Yetter)
Date: Mon, 29 Jun 92 17:28:47 CDT

Surely this must be known, but where is there a reference?

	If one considers any abelian category A, one can from a 2-category
as follows:

	objects: differential graded objects in A (i.e. chain complexes)
	1-arrows: chain maps
	2-arrows: equivalence classes of chain homotopies

where two chain homotopies a,b:F--->G are equivalent if there exits
a map x of degree -2 from the common source of F and G to the common target
of F and G such that

	a - b = xd - dx.

	The two dimensional composition of chain homotopies is addition,
the one dimensional is given by

	if a:F--->G, b:H--->K (source(H)=target(G)) then

	a *1 b = aH + Gb (composites in diagrammatic order).


Also, can one play the game again, and get a 3-category, etc.?

A related question: does anyone know of a notion of "weak n-category" in
which associativities are strict, but the middle-four-interchange 
holds only up to higher dimensional data? (That's what happens here
if one uses chain homotopies instead of equivalence classes.)

--David Yetter
==========================================================================
Subj:	Re: WANTED: Topos Theory
Date: Tue, 30 Jun 92 14:36:15
From: Steve Vickers <sjv@doc.imperial.ac.uk>

Having asked for second-hand copies of Johnstone's "Topos Theory" that I 
could buy, I instead got a couple of commiserational replies saying how 
terrible it is that the book should have been allowed to go out of print.

I intend to write to the publishers, Academic Press, to encourage them to 
reprint in paperback.

If enough people would like to add their "signatures", I'll add a section in 
the following form:

  The following people have told me that they will definitely buy a copy
  if the book is reprinted at not too high a price:

    name               brief address
    name               brief address
    :                  :

Please email me if you'd like to be included.
  
Steve Vickers.

sjv@doc.ic.ac.uk
