Subj:	monad morphisms
From: koslowj@cis.ksu.edu (Juergen Koslowski)
Date: Mon, 3 Feb 92 11:21:32 CST

Defining a morphism <U,f> from an monad <S,e,m> on X
to a monad <S',e',m'> on X' in such a way that U is
a 1-cell (functor) from X to X', there are two choices
for the direction of the 2-cell (nat. trans.) f. Street
in "The formal theory of monads", JPAA 2(1972), 149--168,
chooses f to go from the composit of U with X' to the
composit of X with U, while Barr and Wells in TTT choose the
other direction (although in their case X=X'). The second
choice seems more intuitive to me, since the appropriate
specialization gives ordinary monoid homomorphisms. What is
the rationale for the first choice?

J"urgen Koslowski
Dept. of Math.
Dept. of Comp. & Info. Sci.
Kansas State University
=========================================================================
Subj:	Corrections for  "Categories, Allegories"
Date: Mon, 3 Feb 92 14:40:03 EST
From: pjf@saul.cis.upenn.edu (Peter Freyd)


Aside from some spelling, punctuation and font errors please note
the following corrections to

                      CATEGORIES, ALLEGORIES
                                by
                   Peter Freyd and Andre Scedrov.


1.725 (p 120)

  5'th line down (3'rd equation) should be:

          z ^ (x <--> y)   =   z ^ [(z ^ x) <--> (z ^ y)].

   using  <-->  for the double-arrow operation and  ^  for the meet operation.

1.82(10)  (p 143)

  13'th line down (first in italics) should be:

      A functor that preserves pre-limits preserves limits.

1.947  (p 172)

  4'th line down.  Formula should read

                              R(^F)  <  ^F

  using  ^  for the intersection and  <  for containment.

2.11   (p 196)

  Please note that we begin by saying that an allegory is a category.
  All equations of 1.1 are to be considered part of the definition
  of allegory.

2.158  (p 207-210) small print
                     [thanks to Roger Maddox]

  The sentence about graphs on page 209, 8 lines down, is false:
  "If one identifies any one or any two of the pairs of vertices,
  the resulting graph is not in G-bar."  The trouble is that if the
  vertices labeled  s  and  t  are identified the result is in  G-bar.
  Worse, the displayed formula _is_ a consequence of the allegory
  equations (in which ' is used for reciprication):

             1 ^ (R ^ R')(R ^ R')(S ^ S')(S ^ S )   <    [distributivity]
                   1   4   2   3   2   3   1   4


             1 ^ (R R ^ R'R')(S'S' ^ S S )   =           [2.124]
                   1 2   4 3   2 1    3 4


              Dom((R R ^ R'R') ^ (S'S'^ S S )')  =
                    1 2   4 3      2 1   3 4


              Dom((R R ^ S S) ^ (R R ^ S S )')   =       [2.124]
                    1 2   1 2     3 4   3 4


             1 ^ (R R ^ S S )(R R ^ S S ).
                   1 2   1 2   3 4   3 4


  The subscripts in the complicated formula in the middle of page 210
  are remarkably wrong.  They should be:

                       n-1
  1 ^ (R  ^  R'   )[ INTERSECT (R    ^ R' )(S'   ^ S  )](S' ^ S    )
        0     2n-1    i = 1      2i-1   2i   2i-1   2i    0    2n-1


                   n-1
       <         PRODUCT (R  R    ^ S  S    )
                  i = 0    2i 2i+1   2i 2i+1


  The argument that these formulas are not consequences of the allegory
  equations is OK for  n > 2.

2.357   (p 234)

  9'th line down.  Of the two equations on this line the first
    is, of course, just a restatement of the definition of domain
    of simplicity.  The second, however,  should be referenced
    with  [2.124].

2.4  (p 235)

  5'th line up.  Of the four  R's  two should be  F's.  The
    subscripts should remain  R's.  The "numerators" should become
    F's.

2.412  (p 236)

  12'th line up.  The reference  [2.357] is wrong.  It should be
    [2.124]

2.418  (p 238)  small print

  16'th line up.  The formula  [X]/E  should be  X/I.

2.444  (p 248)  small print

   Not a correction but an improvement.  A nicer example of the
   failure of the law of metonymy is the full sub-allegory of the
   allegory of  Z-sets  (sets with automorphisms) of all those Z-sets
   in which no orbit has more than 3 elements.  For power-objects
   start with the usual construction and remove all orbits with
   more than 3 elements.  If  A  is a 5-element  Z-set consisting
   of two orbits (one with 2 and one with 3 elements) then the epsiloff
   relation from  [A]  to  A  is not tabular.

B.211  (p 272)

    Last sentence.  There are two rules for existential
    quantification and the second is needed for equality.  The
    best correction seems to be simply to remove this last
    sentence (and, of course, the index entry for Horn logic).

B.229  (p 274)

   Not a correction but an addition.  The rules for the commu-
   tivity and idempotence of existential quantifiers are given.
   The same rules for universal quantifiers should also be
   given.

B.3  (p 275-6)

    The definition of DERIVED PREDICATE is correct but too terse.
    Be warned.


corrections for SUBJECT  INDEX

page    Entries to be added:

287     ASSEMBLIES                     2.153
287     CARRIER                        2.153
288     CAUCUSES                       2.153
289    *effective topos                2.418
292     MODULUS                        2.153

Finally, there should not be an asterisk on the index entry for
SUBTERMINATOR on page 295.
=========================================================================
Subj:	Re: monad morphisms
From: Steven John Vickers <sjv@doc.imperial.ac.uk>
Date: Wed, 5 Feb 92 11:24:26 GMT

I believe Street defined both kinds of morphism, calling them "monad functors"
and "monad opfunctors".

The Eilenberg-Moore (category of algebras) construction is functorial with
respect to the functors (this is touched on in MacLane's Categories for the
Working Mathematician, exercise VI.2.3).

The Kleisli construction is functorial with respect to the opfunctors.

Steve Vickers.
=========================================================================
Subj:	Re: monad morphisms
From: street@macadam.mpce.mq.edu.au (Ross Street)
Date: Thu, 6 Feb 92 12:18:00 EST

Defining morphisms between monads in a 2-category C (where C = Cat if you
like) the way I do in "The formal thy of mnds", I obtain a 2-category
Mnd(C) of monads in C such that the inclusion C --> Mnd(C) has the
Eilenberg-Moore construction as its right adjoint. On page 159 of the paper,
J"urgen will find the 2-cell which satisfies his intuition; these are good for
Kleisli's construction.

Of course, we now know that the EM-construction is just a weighted (or
"indexed") limit in the Cat-enriched context ("Limits indexed by cat-valued
2-functors" JPAA 1976?). Kleisli-construction is a weighted colimit. Also
see SLNM 420 (Kelly-Street paper and "Elem cosmoi" Section 6 pp166-168).

Regards,
Ross

=========================================================================
Subj:	Steenrod
Date: Tue, 11 Feb 92 13:15:49 -0500
From: cxm7@po.CWRU.Edu (Colin Mclarty)


I understand Steenrod was interested in axiomatizing homology in the 1940s.  An 
anecdote says that when he first saw category theory he got the idea of 
treating maps in homology on a par with homology groups (rather than as a 
corollary to the construction of homology groups), and he found this was the 
key to axiomatizing the subject.  Does anyone have accurate information on 
this?
=========================================================================
Subj:	Re:  Steenrod
Date: Wed, 12 Feb 92 07:40:22 EST
From: pjf@saul.cis.upenn.edu (Peter Freyd)

In answer to Colin Mclarty's query about Steenrod: one of the first
things he ever told me after he became my dissertation advisor is just
that story.  It had not occured to him to say anything about the action
on maps.  The resulting axiomatization of homology is, of course, his
book with Eilenberg, The Foundations of Algebraic Topology.

He did not deny inventing the phrase "generalized abstract nonsense" to
describe the categorical approach, but he said that it was meant in
the affectionate sense.

=========================================================================
Subj:	Re:  Steenrod
Date: Wed, 12 Feb 92 09:36:45 EST
From: barr@triples.Math.McGill.CA (Michael Barr)

The person to ask is Eilenberg.  I heard Sammy tell the following, though.
he said that Eilenberg had said to him, upon seeing The general theory
of natural transformations, that no paper had ever influenced his
thinking more.  Sammy also repeated that P.A. Smith, a first rate
algebraic topologist of the ``hard'' school had told him that he
had never read a more trivial paper in his life.  Well, maybe it
wasn't quite so strong.  Sammy thought both reactions quite reasonable.

Certainly, Sammy told some such story about Steenrod wanting to axiomatize
homology theory.  My recollection is that Steenrod told him that although
they knew that there was the homology homomorphism induced by a 
continuous map, he had never thought of using that fact as a basis
for his axiomatization.  And that when he read GTNT, the scales fell
from his eyes.  But you should really try to get this story straight
from Sammy before it is too late.

I suspect that one of the obstacles to taking maps seriously is that
the map induced by an inclusion is not an inclusion.  This must have
bothered people quite a lot in those days.  Homomorphism meant surjective
homomorphism and really only subgroups and quotient groups were taken
seriously.  I have often conjectured that were it not for that, 
Birkhoff might have invented categories instead of lattices.

Michael
=========================================================================
Subj:	Algebraically complete categories
Date:         Wed, 12 Feb 92 16:01+0000
From:         mxh%dcs.edinburgh.ac.UK@QUCDN.QueensU.CA


Is it known which functions from N to N are representable in an
algebraically closed ccc? (A ccc in which initial algebras for "all"
endofunctors exist. "all" includes at least those definable by x and
=> and those arising from initial algebras like e.g. the List functor.
More precisely are there functions representable in F, which aren't in
an algebraically closed category?

-Martin Hofmann
=========================================================================
Subj:	Re:  Algebraically Complete Categories
Date: Thu, 13 Feb 92 09:08:30 +0100
From: dybkjaer@euler.ruc.dk (Hans Dybkjaer)

A simply typed lambda-calculus with types x, =>, and natural numbers can
express functions N->N equivalently to those provably total in first order
Peano arithmetic (cf. [Girard 89]). This encompass, e.g., the Ackermann
function. 

Though I do not remember any reference, I do not think that more expressive
power in terms of functions N->N is gained by adding other initial algebras,
like List.

You might also be interested in Tatsuya Hagino's categorical programming
language CPL ([Hagino 87][Wraith 89][Dybkj{\ae}r 91]) which is based on
(restricted) initial and final dialgebras (a dialgebra is an arrow f:
F(A)->G(A) where F,G:C->D are functors). Dialgebras make it possible to define
initial and final algebras, products, coproducts, and exponentials from
scratch. 


---Hans Dybkj{\ae}r


Dybkj{\ae}r, Hans [1991]: "Category Theory, Types, and Programming Languages",
	PhD thesis, DIKU, University of Copenhagen, May 1991. Tecnical report
	91/11.

Girard, Jean-Yves, Yves Lafont, and Paul Taylor [1989]: "Proofs and Types",
	Cambridge University Press.

Hagino, Tatsuya [1987]: "A Categorical Programming Language", PhD thesis, LFCS,
	University of Edinburgh, September 1987. Technical report CST-47-87.

Wraith, Gavin C. [1989]: "A Note on Categorical Datatypes", in "Category Theory
	and Computer Science, Manchester UK, September 1989", Proceedings,
	edited by David H. Pitt et al., Springer-Verlag, LNCS 389.
=========================================================================
Subj:	modified realizability topos and so on
Date: Thu, 13 Feb 92 17:21:17 MET
From: Thomas Streicher <streiche@fmi.uni-passau.de>


It is well known that one can build toposes from realizability.
This has been done quite successfully for Kleene realizability and  
for extensional realizability.
An excellent survey on these topics is contained in the Thesis by  
Jaap van Oosten.

There he also suggests how to give higher order variant of MODIFIED  
REALIZABILITY - although only of that variant of modified  
realizability called the "HRO variant" in Troelstras book (SLNM 344).

I want to suggest an - I think more adequate - version.

Jaaps approach :  propositions are modelled as as pairs  (p,P)
                  where  p  is a subset of  P  which is a set of 

                  natural numbers containing  0

                  (p,P) |- (q,Q)   ("entailment")

                  iff there is a natural number  n  such that
               

                    for all m in P :  {n}(m) terminates  and 

          

                                      {n}(m) is in Q   and

                                      {n}(m) is in q  if  m  is in p 


It is felt - and also explicitely expressed in Jaaps Thesis - that  
the requirement " 0  in  P " is a very liberal way of expressing that  
the set P  of  "potential realizers"  is NOT EMPTY.

The intuition behind modified realizability is much stronger :

a proposition should be a pair  (D,P)   where  D  is a  "domain of  
potential realizers"  and  P  is an arbitrary subset of  D .
Of course, the requirement that  D  is a domain is much stronger than  
the claim that it contains a specified element (namely 0).

Now my suggestion is that 


PROPOSITIONS are  pairs  (D,P)  where  D  is an effective domain    
and  P  is an arbitrary subset of  D .
By "domain"  we either mean a complete Sigma extensional per or a  
Sigma replete per with a least element (realized by the code of the  
totally undefined function from N to N ).

Now one does know that families of pers form a hyperdoctrine over SET  
and so do families of domains in the sense specified above.

NOW THE POINT IS that this holds also for families of pers with a  
predicate on it.
We give the skeleton of a definition below.

We construct the following posetal hyperdoctrine over  SET .

Let DOM be any of the subsets of PER mentioned above.

An object over  X  is a pair  (F,P)  where  F : X -> DOM  and  P   
associates with any  x  in  X  a subset  P(x)  of  F(x).

Now if  (F,P)  and  (G,Q)  are objects over  X  then  (F,P) |- (G,Q)   
iff there is a natural number  n  such that

  -  for any  x  in  X :  n realizes a morphism f(x) : F(x) -> G(x)

  -  for all x in  X  and  d  in  P(x) :  f(x)(d)  is in  Q(x)   .

Obviously, any fibre is a pre-Heyting algebra and it is complete in  
the sense that right adjoints to reindexing exist and are themselves  
preserved by reindexing.

We also have a generic element over

  Prop := { (D,P) |  D  in  DOM  and   P  is a subset of  D }

where the generic object in the fibre is given by  (F_gen,P-Gen) :

  F_gen(D,P) = D    and    P_gen(D,P) = P  .

Now from this MODIFIED REALIZABILITY TRIPOS one can construct the  
associated  MODIFIED REALIZABILITY TOPOS .

It remains as an open problem whether it is very different from the  
one Jaap proposed.

REMARKS  One could as well consider the triposes arising from 

         families of predomains or families of extensional pers.

         Our considerations were directed towards 

         domains of partial elements.

         Of course it is also possible to consider the collection of 

         domains of total elements, e.g. all Delta replete pers
         (where Delta is the 2-element discrete per) 

         s.t. any constant total function with value  0  or  1  

         realizes some element .

    QUITE GENERALLY ANY SUBCOLLECTION OF PERS CLOSED UNDER PRODUCTS 

    OF FAMILIES OVER UNIFORM omega-Sets  ALLOWS TO DEFINE A   

    CORRESPONDING REALIZABILITY TRIPOS and ITS ASSOCIATED 

    REALIZABILITY TOPOS.


P.S.  Maybe all these facts are already well known to Martin Hyland !   

      In his Como paper he says that he has checked several topoi 

      arising from several versions of realizability or functional   

      interpretation in form of "back-side of an envelope 

      computations" .
      Nevertheless the remarks above - I hope - might be interesting 

      for some people (as e.g. me) who have not had such a close look 

      at the backside of Martins envelopes.

   For toposes derived from functional interpretation at first sight 

   there is the problem that Dialectica categories are not ccc-s but 

   categorical structures suitable for interpreting classical linear  

   logic. But if one looks at the Kleisli coalgebra associated with 

   the  !-comonad  on ecan again get a tripos corresponding to 

   functional interpretation !


Thomas Streicher
=========================================================================
Subj:	Q-construction  
Date: Thu, 13 Feb 92 16:35:13 EST
From: vladimir@math.harvard.edu (Vladimir Voevodsky)

Notice that Quillen's Q-construction can be carried out in a non-abelian 
category as well. Does anyone know what kind of classifying space one 
gets applying Q-construction to an arbitrary topos? 
( The answer for Sets is well-known and nontrivial: stable 
homotopy type of spheres ). Thanks in advance.
=========================================================================
Subj:	Category Theory at the Isle of Thorns
Date: Thu, 13 Feb 92 15:48 GMT
From: MMFC6@cluster.sussex.ac.uk

                     CATEGORY THEORY AT THE ISLE OF THORNS
                          12TH JULY - 17TH JULY 1992


The University of Sussex invites you to attend the Sixth International Meeting
on Category Theory and its Applications to be held at the White House, the Isle
of Thorns, Chelwood Gate, East Sussex, England, from Sunday, 12th July to
Friday, 17th July, 1992.  The Isle of Thorns is the Conference Centre of the
University of Sussex, situated in the Ashdown Forest, about 25 miles north of
Brighton.

The meeting follows the Joint Meeting of the American Mathematical Society and
the London Mathematical Society to be held in Cambridge, England from Monday,
29th June to Wednesday, 1st July, 1992, and the European Congress of
Mathematicians to be held in Paris, France from Monday, 6th July to Friday,
10th July 1992.

In order to receive further details of the meeting, together with an
application form for accommodation at the meeting, please reply by sending your
name and postal address either electronically to
c.j.mulvey@cluster.sussex.ac.uk, or postally to:

                    Dr. Christopher J. Mulvey,
                    Mathematics Division,
                    University of Sussex,
                    Falmer,
                    BRIGHTON, BN1 9QH,
                    United Kingdom.

Any electronic request received will be acknowledged electronically, and the
details sent by postal mail.  If you do not receive this acknowledgement, your
message can be assumed not to have arrived.  In recent times, this has been a
not infrequent occurrence.

The completed application form must be returned by the closing date of 1st May,
1992.  

It would be appreciated if you would forward a copy of this notice to anyone
who might be interested in receiving it who might otherwise not receive it.



                                             Christopher Mulvey.iv
=========================================================================
Subj:	geometric functiors
Date: Fri, 14 Feb 92 16:27:19 -0500
From: cfw2@po.CWRU.Edu (Charles F. Wells)


I would like to know more about geometric functors whose inverse
images preserve universal quantifiers.  
I've been told Mikkelson has studied something like these in
connection with compactification.  Anyone know a reference?

=========================================================================
Subj:	New address
Date: Sat, 15 Feb 92 13:58:41 EST
From: barr@fermat.Math.McGill.CA (Michael Barr)

As a temporary measure, all mail to triples.math.mcgill.ca should now
be sent to fermat.math.mcgill.ca.
In the future we will probably install a dept server and use the address
math.mcgill.ca for all.  This will avoid a disaster when a machine goes
kaput.  What happened is a long story, but I have bought my last Sun
computer.  I'd sooner do business with a used car salesman.

Michael
=========================================================================
Subj:	Algebraically Complete Categories
Date:         Sun, 16 Feb 92 15:27+0000
From:         mxh%dcs.edinburgh.ac.UK@QUCDN.QueensU.CA

In "Re:  Algebraically Complete Categories" dybkjaer@euler.ruc.dk
(Hans Dybkjaer) writes:

    Though I do not remember any reference, I do not think that more expressive
    power in terms of functions N->N is gained by adding other initial algebras
,
    like List.

If we restrict our attention to initial algebras for a *polynomial*
functor this is true, but in fact if we have an initial algebra for
the functor T(X)=1+X+(N=>X) (Coquand-Huet's ordinal notations) we can
express the functions $F_{\epsilon_0}$ [1] and even $F_{\Gamma_0}$ [2],
where F_ denotes the fast growing hierarchy. These functions are not
expressible in G"odel's T. I conjecture that algebraically complete
categories are at least as expressive as system F.

-- Martin Hofmann

References:

[1] T. Coquand and C. Paulin-Mohring: "Inductively defined
     types" in LNCS 389

[2] J. Gallier: "What's so special about Kruskal's theorem and the
ordinal $\Gamma_0$" in Annals of pure and applied logic (July 1991)
=========================================================================
Subj:	Re:  Algebraically Complete Categories
Date: Sun, 16 Feb 92 12:33:39 EST
From: pjf@saul.cis.upenn.edu (Peter Freyd)

Martin Hofmann asks:

  Is it known which functions from N to N are representable in an
  algebraically closed ccc? (A ccc in which initial algebras for "all"
  endofunctors exist.)

I don't know about algebraically closed ccc's but I should have answered
this question in my Durham paper for the algebraically compact case.
(Algebraically compact means every endofunctor has both an initial algebra
and a final coalgebra and they are canonically isomorphic.)  The
only algebraically compact ccc is the degenerated category, so we ask
for it not to be a ccc but to be a reflective subcategory of an ambient ccc.
To get off the ground for a (flat) natural numbers object, N, we ask that 
the algebraically compact category have finite coproducts.  Then the answer
is, essentially, that every recursive function appears as an endomorphism 
of  N.  To make this precise, define a "point" of  N  to be a map thereto
from the terminator, a "standard point" to be either the  0-point or one of
its successors.  Any endomorphism on  N  induces a partial endomorphism
on the standard points.  The result is that every recursive partial 
function is induced by an endomorphism on  N.

Since there is something of a free structure for this theory we can
not expect to get more than the recursive.

To return to the original question: does someone have a counterexample
to back up Hans Dybkjaer's comment:

  Though I do not remember any reference, I do not think that more expressive
  power in terms of functions N->N is gained by adding other initial algebras
=========================================================================
Subj:	regular monos, epis and categories
From: Paul Taylor <pt@doc.imperial.ac.uk>
Date: Mon, 17 Feb 92 14:50:58 GMT

I am a bit confused about what is or should be the correct definitions of
regular monos, epis and categories.  In the traditional examples from
algebra there are plenty of limits and colimits around, so it doesn't
much matter whether you say they're the (co)equalisers of arbitrary pairs
or their (co)kernel pairs.

Has anyone made use of these concepts in categories which have only the
bare minimum (whatever that may be) of limits and colimits?

Paul Taylor.
=========================================================================
Subj:	Re: Algebraically Complete Categories
Date: Mon, 17 Feb 92 17:51:54 MET
From: Thomas Streicher <streiche@fmi.uni-passau.de>

Peter Freyd claims that any partial recursive function arises as  
Hom(1,f)  for some endomorphism from  N  to  N . 

From his comment I could not see why.
If one assumes the solution

      D = N + [N->N]

then one may define the recursor  Y  in the usual lambda calculus  
style.

BUT how can one transform the exponentiation functor in the ambient  
ccc to a functor on the reflective cat of strict maps ?

I guess one needs a little bit more assumptions :

e.g. that the category of predomains contains the reflective subcat  
of domains and strict maps which in turn contains the category of  
total strict maps (by the lift functor) which is equivalent to the  
category of predomains .

Can one axiomatize this situation in a sufficiently strong way, e.g.   
that one can transfer the exponentiation functor from the category of  
predomains to the category of domains and strict maps ?

Thomas Streicher
=========================================================================
Subj:	jobad
Date: Tue, 18 Feb 92 9:17 GMT
From: MAS010@vaxa.bangor.ac.uk

UNIVERSITY OF WALES FELLOWSHIP
Applications are invited for a Fellowship for two years from 
October 1, 1992, open to graduates of any University, and tenable 
in the School of Mathematics, the University of Wales, Bangor.   
The Fellowship is primarily for research, and candidates should 
possess a research degree.  Preference is likely 
to be given to candidates with expertise related to one or more of 
Algebraic Topology, Category Theory, Theoretical Computer Science.

The stipend will normally be (sterling) 12,129 (first year) rising to 
12,860 (second year) (under review).

The Fellowship is sponsored by the University's Validation Board, 
and further details and application forms may be obtained from the 
Secretary, University of Wales Validation Board, 22 Park Place, 
Cardiff CF1 3DQ, United Kingdom, to whom completed forms should be 
returned not later than March 20, 1992. 
FAX:  International (+44)-222-230820    UK:  (0222) 230820
EMAIL for School of Mathematics, Bangor.:
MAS006@UK.AC.BANGOR
FAX for School of Mathematics, Bangor: (248)355881
=========================================================================
Subj:	Re:  regular monos, epis and categories
Date: Tue, 18 Feb 92 08:59:29 EST
From: barr@fermat.Math.McGill.CA (Michael Barr)
Here is what I think ought to be the definition in a general category.
Say that f:A --> B is a regular epi if whenever g:A --> C is an arrow with
the property that [(forall parallel pairs u,v with codmain A)
(fu = fv ==> gu = gv)] ==> (exists! h:B --> C) (hf = g).  If there is
a kernel pair, this is equivalent to the more common definition.
Obviously, the dual definition should be used for regular monics.

Michael
=========================================================================
Subj:	Re:  regular monos, epis and categories
Date: Tue, 18 Feb 92 09:23:48 EST
From: pjf@saul.cis.upenn.edu (Peter Freyd)

Paul Taylor asks for the "correct" definition of regular monos
and epis.  I think a regular epi has always been an epi that 
appears as a coequalizer.  The lemma is that if an epi has a pullback
with itself then it is regular iff it is the coequalizer of the
relevant pair of maps from that pullback (aka the "kernel pair", the
"level", the "congruence").  But I can't imagine anyone insisting 
that split epis are regular only when they have kernel pairs.
=========================================================================
Subj:	Re: Algebraically Complete Categories
Date: Tue, 18 Feb 92 09:35:59 EST
From: pjf@saul.cis.upenn.edu (Peter Freyd)

Thomas Streicher wrote:

>Peter Freyd claims that any partial recursive function arises as  
>Hom(1,f)  for some endomorphism from  N  to  N . 

>From his comment I could not see why.
>If one assumes the solution

>      D = N + [N->N]

>then one may define the recursor  Y  in the usual lambda calculus  
>style.

>BUT how can one transform the exponentiation functor in the ambient  
>ccc to a functor on the reflective cat of strict maps ?


Actually, I did not make that claim since  Hom(1,f)  can't be recursive
if  Hom(1,N)  is, for example, an uncountably infinite set.  What I said
was:

 I don't know about algebraically closed ccc's but I should have answered
 this question in my Durham paper for the algebraically compact case.
 (Algebraically compact means every endofunctor has both an initial algebra
 and a final coalgebra and they are canonically isomorphic.)  The
 only algebraically compact ccc is the degenerated category, so we ask
 for it not to be a ccc but to be a reflective subcategory of an ambient ccc.
 To get off the ground for a (flat) natural numbers object, N, we ask that 
 the algebraically compact category have finite coproducts.  Then the answer
 is, essentially, that every recursive function appears as an endomorphism 
 of  N.  To make this precise, define a "point" of  N  to be a map thereto
 from the terminator, a "standard point" to be either the  0-point or one of
 its successors.  Any endomorphism on  N  induces a partial endomorphism
 on the standard points.  The result is that every recursive partial 
 function is induced by an endomorphism on  N.

There is a missing word, I'm afraid, in that paragraph.  The assumption
should be that the algebraically compact category is a reflective _lluf_
subcategory of a ccc (that is, the two categories have the same objects).
If that word is left out, we are stuck with the fact that the full
subcategory of the terminator is always a reflective algebraically compact
subcategory (of any category with a terminator).   

Let  S  be the reflection of the terminator (I denoted it with sigma
in the Como and Durham papers).  Let  N  be the initial algebra for the
functor \X. S+X  ( "\" for lambda,  + for the coproduct in the algebraically
compact subcategory, denoted as "wedge" in the Como/Durham papers).  The 
result stands as now stated, where  0:1 -> N  is the composition

    1  ->  S  ->  S+N  ->  N     (the first arrow is the reflector, the
                              second is from the coproduct structure and
                              the third is the algebra structure on  N)

and the successor map  N -> N  is 

    N  ->  S+N  ->  N     (the first arrow is from the coproduct structure,
                            and the second is the algebra structure on  N).

The proof requires the existence a fixed-point operator, that is, a 
dinatural transformation from  \X.(X=>X)  to  \X.X.  A proof is in the 
Durham paper, which paper contains, I submit, a positive answer to 
Streicher's question:

>Can one axiomatize this situation in a sufficiently strong way, e.g.   
>that one can transfer the exponentiation functor from the category of  
>predomains to the category of domains and strict maps ?

The answer is the theory of algebraically compact categories with
finite coproducts which appear as reflective lluf subcategories of ccc's.
=========================================================================
Subj:	Re: regular monos, epis and categories
From: Paul Taylor <pt@doc.imperial.ac.uk>
Date: Tue, 18 Feb 92 21:21:34 GMT

Much as I try to get out of the habit of finding counterexamples to things,
here, for what it's worth, is a category with a map satisfying Mike Barr's
condition but which is not a coequaliser. 

        * ====> * <==== *
        | \   / | \   / |  
        |  \ /  f  \ /  |
        |   X   |   X   |
        |  / \  |  / \  |
        V V   V V V   V V
        *       *       *

or, more prettily,
  $$\begin{diagram}
    \bullet&\pile{\rArr\\\rArr}&A&\pile{\rArr\\\rArr}&\bullet\    \dArr&\SE\SW&\dArr~f&\SE\SW&\dArr\    \bullet&&B&&\bullet
  \end{diagram}$$

("=" above means a parallel pair. Believe it or not, I typed that without
an editor!)

A mono satisfying Mike's condition is invertible.  Can't see how to relate
it to orthogonality (in the sense of factorisation systems) with monos.

What I had in mind was that someone may have used coequalisers in distributive
categories to model "while".  Only those coequalisers which, as relations and
hence directed graphs (the node set being the target of the parallel pair),
are directed arise in this way.

Paul Taylor
=========================================================================
Subj:	Re:  regular monos, epis and categories
Date: Wed, 19 Feb 92 12:38:36 EST
From: pjf@saul.cis.upenn.edu (Peter Freyd)

To make my remark, ("I think a regular epi has always been an epi 
that appears as a coequalizer") compatible with Mike Barr's proposed
definition just understand the word "coequalizer" to allow the case of 
a (joint) coequalizer of a family of pairs of maps.
=========================================================================
Subj:	Re:  regular monos, epis and categories
Date: Wed, 19 Feb 92 16:28:11 EST
From: barr@fermat.Math.McGill.CA (Michael Barr)

What Peter says about coequalizers of a family of pairs is consistent
with what I said so long as you allow a family to be large.  As I wrote
to PT earlier, these epics are strict, orthogonal to all monics and 
are coequalizers of their kernel pairs, if such exist.  --Michael
=========================================================================
Subj:	Re:  regular monos, epis and categories
From: street@macadam.mpce.mq.edu.au (Ross Street)
Date: Fri, 21 Feb 92 10:01:47 EST

My message related to Paul Taylor's question seems not to have been received
since it has not yet appeared on the Bulletin Board:

I recommend the following two papers in answer to this question of Paul Taylor:

G.M. Kelly, Monomorphisms, epimorphisms, and pullbacks, J. Australian Math Soc
Volume IX (1969) 124-142

P. Freyd and G.M. Kelly, Categories of continuous functors I, J. Pure Appl
Algebra Volume 2 #3 (1972) 169-191

Regards,
--Ross Street

In any case, these references are also consistent with the comments of Mike
Barr and Peter Freyd.

--Ross
=========================================================================
Subj:	Paul Taylor's question
Date: Fri, 21 Feb 92 23:52:16 +1100
From: kelly_m@maths.su.oz.au (Max Kelly)

I thought I had sent an answer to Paul Taylor's question,
but - since I have not seen it on the bulletin board - I
suppose I slipped up somewhere in transmitting it. Anyway,
as Ross Street has pointed out in his message of today, the
whole matter was discussed in detail in my old paper "Mono-
morphisms, epimorphisms, and pull-backs", J.Austral. Math. Soc.
9(1969),124-142 - which contains some interesting results,
besides being couched in the generality that Paul seeks. Were
I writing it today, I should change it in places; certainly,
I should speak of ARBITRARY intersections of monos, or of
strong monos, or of regular monos, rather than of SMALL
intersections coupled with WELLPOWEREDNESS. There are good
things in Freyd-Kelly, "Categories of continuous functors I",
J.Pure Appl. Algebra 2(1972),169-191,but we chose there to
speak not of STRONG epis but of EXTREMAL ones, these coin-
ciding under our hypotheses there. In several later papers
I have refined various points from that very early paper -
but it contains the guts of the matter.

                           Max Kelly, 21 Feb.
=========================================================================
Subj:	The ``monoid'' of endofunctors of some categories
Date:    Wed, 26 Feb 1992 12:05:01 -0500 (EST)
From:    D_FELDMAN@UNHH.UNH.EDU

I would like to pose a question, first in the concrete situation where it
arose, and then more abstractly.

Let  n-MAN  be the category of (topological) n-manifolds and continuous
maps.  My question is, what are the endofunctors of  n-MAN ?  

To start with, there is no shortage.  Any functor
                         F: n-MAN --> Set  
gives rise to an endofunctor F': n-MAN --> n-MAN  as follows.
On objects  M \in n-MAN ,   F'(M)=M x F(M) (where F(M) is given the discrete
topology) and for morphisms f \in n-MAN ,  F'(f)(p,x)=(p,f(x)) .
Let us call these endofunctors of set type.
The endofunctors of set type form a proper class, since for any
cardinal k  there is a set valued functor taking M to the underlying
set of  M x k.  So one may ask if there are any others?  If there are,
then one can ask a more refined question.
The endofunctors of n-MAN form a (proper class based) monoid. 
It still makes sense to consider the monoid ideal generated by the class of
endofunctors of set type.  One might then try to form the
quotient monoid and as a measure of how close the class of set type 
endofunctors comes to exhausting the class of all endofunctors.

Is this construction familiar?  If one considers instead the category
of pointed n-manifolds, then one has the universal covering space endofunctor,
which is not of set type; there the quotient monoid will be non-trivial.

A few more thoughts:
The construction above makes sense in any category with arbitrary
coproducts, or even just copowers, if that is the right term.

If instead of the category of n-manifolds, I had considered the
category of connected n-manifolds (and continuous maps), then the
set type endofunctors are no longer available - so is the monoid
of endomorphism of this category related the quotient monoid alluded
to above (same question for pointed connected n-manifolds.)

In the case of the category of n-manifolds (but not pointed n-manifolds)
there is a large supply of constant maps.  Given an n-manifold M,
the lattice of subobjects of M.  Fixing another manifold N, consider all
the constant maps from N to M or from N to a subobject of N.  If an endofunctor
F preserves constant maps (and nontrivial endofunctors of set type don't)
then considering all this structure, I don't expect that it will be
difficult to show that F is trivial.

I would appreciate any reactions or references to similar considerations
in the literature.  Thanks,

					David Feldman
					University of New Hampshire
=========================================================================
Subj:	Union Conference
Date: 26 Feb 92 15:07:00 EDT
From: "NIEFIELD, SUSAN" <niefiels@gar.union.edu>


UNION  COLLEGE  MATHEMATICS  CONFERENCES   April 25 and 26, 1992 

CATEGORY THEORY    DYNAMICAL SYSTEMS    NUMBER THEORY

	Keynote Speaker		JOHN MILNOR

The Mathematics Department of Union College is pleased to announce
its eighth occasional spring conference.  The meeting will feature
a keynote address by John Milnor, as well as parallel sessions in
Category Theory, Dynamical Systems, and Number Theory.

-------------------------------------------------------------

Invited Speakers for the Category Theory section:

	       F. WILLIAM LAWVERE  and  JOAN PELLETIER

Invited Speakers for the Number Theory Section:

  KARL RUBIN  "Rational points on elliptic curves with complex multiplication"

  JOSEPH SILVERMAN  "Canonical heights on varieties with morphisms"

Invited Speakers for the Dynamical Systems section:

                           To be announced

-------------------------------------------------------------

			  CONTRIBUTED TALKS

Please contact one of the organizers of the parallel session in which you 
would like to give a talk.  Such talks should be 20-30 minutes long.

Conference Organizers for Category Theory:

  Susan Niefield     niefiels@union.bitnet  or  niefiels@gar.union.edu
  Kimmo Rosenthal    rosenthk@union.bitnet  or  rosenthk@gar.union.edu

Conference Organizer for Dynamical Systems

  Michael Frame        framem@union.bitnet  or  framem@gar.union.edu

Conference Organizers for Number Theory:

  William F. Hammond  (SUNY Albany)             hammond@csc.albany.edu
  Karl Zimmermann    zimmermk@union.bitnet  or  zimmermk@gar.union.edu

Department of Mathematics, Union College, Schenectady, NY  12308-2311
	    Telephone (518) 370-6246   FAX (518) 370-6789

-------------------------------------------------------------

		   DETAILED CONFERENCE INFORMATION

Department of Mathematics, Union College, Schenectady, NY  12308-2311
	    Telephone (518) 370-6246   FAX (518) 370-6789

Each session will include invited lectures and shorter contributed talks.  
Your $25 registration fee will cover the cost of a reception Friday night, 
and a cocktail hour, buffet dinner, and party on Saturday.  The fee for 
graduate students is $15.

--------------------------------------------------------------

			    ACCOMMODATIONS

To obtain a room at one of the rates listed below, mention Union College 
when you call.  A block of rooms has been set aside at the Days Inn.  All 
of the hotels are within easy walking distance of campus.

1.   Days Inn     (518) 370-0851
     $49 single       $49 double
     (includes continental breakfast)

2.   Holiday Inn  (518) 393-4141
     $58 single       $63 double

3.   Ramada Inn   (518) 370-7151
     $60 single       $70 double

----------------------------------------------------------------

			       SCHEDULE

Friday Evening, April 24

 9:00-11:00  Reception  /  2nd floor, Bailey Hall

Saturday, April 25

 9:30-10:00  Registration, coffee & donuts  /  2nd floor, Bailey Hall
10:00-12:00  Parallel sessions  /  2nd floor, Bailey Hall
12:00- 1:30  Lunch  /  Hale House
 1:30- 3:30  Parallel sessions  /  2nd floor, Bailey Hall
 3:30- 4:00  Coffee  /  Social Sciences 016
 4:00- 5:00  Keynote Address by John Milnor  /  Social Sciences 016
 5:30- 6:30  Cocktails  /  Hale House Lower Lounge
 6:30- 8:00  Buffet Dinner  /  Hale House
 8:00-11:00  Dessert and Party  /  Milano Lounge

Sunday, April 26

 9:00- 9:30  Coffee and donuts  /  2nd floor, Bailey Hall
 9:30-12:00  Parallel sessions  /  2nd floor, Bailey Hall
12:00- 1:30  Lunch  /  Hale House
 1:30- 3:30  Parallel sessions  /  2nd floor, Bailey Hall

The parallel sessions will take place in Bailey Hall, 2nd floor.
A schedule and room assignments will be posted outside the Math lounge.

For more information contact one of the organizers or contact:

Department of Mathematics, Union College, Schenectady, NY  12308-2311
	    Telephone (518) 370-6246   FAX (518) 370-6789

------------------------------------------------------------

		       REGISTRATION INFORMATION

	 Registration Fee:  $25    (Graduate Students:  $15)

Participants are urged to register in advance so that necessary arrangements 
can be made.  If this is impossible, you may register on Saturday morning at 
the conference, but please contact us in advance via e-mail or phone.

If you have any questions or need assistance with your travel arrangements, 
please call one of the organizers at (518) 370-6246.

------------------------------------------------------------

			  REGISTRATION FORM

For advance registration please send this information by any convenient method 
to one of the organizers or to the Mathematics Department, Union College at 
the address above.


         Name:

  Institution:

      Address:

         City:

        State:

         Zip :

        Phone:

Email address:

   Check here:     if you are interested in giving a contributed talk.

            Session:

              Title:

=========================================================================
Subj:	Temporary Lectureships
Date:        Thu, 27 Feb 92 13:55:59 AST
From:        <mikef%dcs.ed.ac.uk@UNBMVS1.csd.unb.ca>

To: mikef
Subject: Temporary Lectureships
Date: Thu, 27 Feb 92 17:22:57 GMT
From: Michael Fourman <mikef@ed.dcs>


Please circulate the following announcement,
apologies to those who have already seen this elsewhere.

We hope to make *two* appointments.

- - ---- Start of forwarded text ----
> Received: from gruna.dcs.ed.ac.uk by dcs.ed.ac.uk id aa00780;
>           30 Jan 92 16:02 GMT
> Date: Thu, 30 Jan 92 16:02:01 GMT
> Message-Id: <16831.9201301602@gruna.dcs.ed.ac.uk>
> From: h o d <hod@uk.ac.ed.dcs>
> Subject: Advert
> To: lmp@uk.ac.ed.dcs
> Status: RO
>
>
>                            University of Edinburgh
>                         Department of Computer Science
>
>
>                             Temporary Lectureship
>
> Applications are invited for a temporary (five year) lectureship available
> from October 1992. Applicants should be qualified to Ph.D. level and should
> be able to teach across a range of topics and levels (including first-year)
> within the subject.
>
> The successful candidate will be expected to make a strong contribution to
> research in the Department which currently has major interests in parallel
> computing through its links with the Edinburgh Parallel Computing Centre
> (EPCC) and in theoretical topics through the work of the Laboratory for
> Foundations of Computer Science (LFCS). Candidates with excellent records
> in other areas are also encouraged to apply.
>
> The Department has a very high reputation for the quality of both its
> teaching and research, and has excellent facilities which include over
> 200 workstations and access to a Connection Machine and a Meiko Computing
> Surface in EPCC. The existing staff complement consists of 28 lecturing
> staff (including 5 professors) and over 20 research workers, supported by
> computing officers and technical and secretarial staff.
>
> Initial salary will be on the Lecturer A scale   #12,860 - 17,827
>
> with placement according to age, qualifications and experience.
>
> Further particulars may be obtained by writing to:
>
>     The Personnel Office
>     University of Edinburgh
>     1 Roxburgh Street
>     Edinburgh  EH8 9TB
>
> to whom applications should be sent before the closing date of 31 March 1992,
> or by e-mail from Ms Laura Paterson <lmp@dcs.ed.co.uk>.
>
>
- - ---- End of forwarded text ----

- ------- End of Forwarded Message

=========================================================================
Subj:	Re: The ``monoid endofunctors of some categories
From: dyetter@math.ksu.edu (David Yetter)
Date: Thu, 27 Feb 92 15:50:23 CST

By way of comment on David Feldman's question. One can trivially produce
lots of endofunctors of n-MAN. Since n-MAN had coproducts, so does
any functor category targetted in n-MAN, in particular END(n-MAN).
Second, note that given any endofunctor and a functor from n-MAN to
SET, one can take a copower of the endofunctor by the set-valued
functor. (Feldman's "functors of set-type" being an example of this
in the case where the endofunctor is the identity, but one could
just as easily take say X |----> pi_0(X) x A for a fixed n-manifold, A.)

It might be better to stick to connected n-manifolds, in which case
I'm not sure how to construct any endofunctors other than the constant
ones and the identity.

---David Yetter
=========================================================================
Subj:	alternative name for stable functors
From: Paul Taylor <pt@doc.imperial.ac.uk>
Date: Wed, 26 Feb 92 22:30:14 GMT

A stable functor has a left adjoint on each slice (multiajoint a` gauche,
to quote Yves Diers).  The word stable was first used in this sense by
Gerard Berry (I believe), but it's most unfortunate as it already has
at least one generic meaning in category theory, namely preservation by
pullbacks.

Can anyone suggest a better name?  Here are some possibilities.

1) Stable endofunctors of the category of sets have power series expansions, so
	"analytic"
(after Joyal) is a possibility. It goes nicely with continuous (in the sense of
Dana Scott) although I don't want it to include preservation of filtered
colimits.  Also, as used by Andre' Joyal and Franc,ois Lamarche, analytic
functors need only send pullbacks to weak pullbacks.

2) My definition of a functor U:M->C being stable is that every map X->UC
in C factor as a *candidate* (diagonally universal map in Diers' terminology)
followed by Uh for some h:A->B in M. cf factorisation systems.
This suggests
	"quotate" or "democratic"
(ie there are enough candidates).

3) Much of my work on stable functors as morphisms of domains is based on
a technique of considering slices. This suggests
	"laminated functor"
and similarly laminated category, laminated ccc, laminated topos for the
corresponding well-behaved categories (the last for a category every slice
of which is a topos, not necessarily with a terminal object).

4) Since they're functors with (non-chosen) adjoints on each slice,
	"slice-adjunctible"
is another possibility.

5) I'm also interested in functors which just preserve (binary) pullbacks
(between categories with them).  I'd like to use the word "cartesian" to
refer to anything concerned with pullback squares (and not just products),
and call a pullback-preserving functor
	"cartesian"

Any thoughts?

Paul Taylor

=========================================================================
Subj:	New home telephone number for Max Kelly
Date: Fri, 28 Feb 92 16:25:34 +11
From: kelly_m@maths.su.oz.au (Max Kelly)

From Mon 2 Mar this will be: (-61-2)- 983-9985.
 Regards to all, Max - 28 Feb.
