AN ONGOING COLLABORATION

by Guillaume Brümmer

The mathematician who
has read the preceding "Citation for the award of an honorary DSc to Professor
Bernhard Banaschewski"* *will have noted that the present story starts
in the northern summer of 1975 at a Category Theory meeting in the Black
Forest, and may have guessed correctly that the venue was the *Mathematisches
Forschungsinstitut Oberwolfach*.From the records: Banaschewski's talk
there was on "Categorial (sic) aspects of atomic compactness of structures".

The auspicious moment to invite him to UCT occurred two weeks later at a party at Horst Herrlich's house, which was at that time on the idyllic dyke¹ Oberblockland near Bremen.

The event at UCT, in
August 1976, was the *Symposium on Categorical Topology*, organised
by the Topology Research Group, chaired by Keith Hardie (the founder of
our group, best known for his contributions to homotopy theory). The participants
from abroad were Banaschewski, Herrlich, Lamar Bentley, Peter Collins,
Buzz Fay, Darrell Hajek, Martin Harvey, Gavin Hitchcock, Kazumi Nakano,
Louis Nel, John Rutter, Joachim Schröder, Walter Tholen and Barbara
Smith Thomas.

Naming all these people,
I recall the pervasive enthusiasm over a fresh young field of mathematics
while we, the locals, felt gratified but also perplexed by the momentary
intermission in our isolation. Many of the visitors from then on
maintained close connections with South African mathematics. Herrlich had
already participated in a small UCT symposium in 1974. For Banaschewski
we had arranged a tour of several universities around the country prior
to the symposium. Upon his return to McMaster he wrote me that his visit
had been of "the kind that gives one a new vision of human affairs. ...
I have passionately fallen in love with South Africa".

The South African Mathematical
Society (SAMS) launched its journal *Quaestiones Mathematicae *(QM)
in November 1976. Horst Herrlich was one of the initial nine members of
the Editorial Board of QM. The refereed proceedings of our 1976 Symposium
were published in QM, volume **2**, parts 1-3, (1977), 432 pages, edited
by Hardie and myself. The issue contained Banaschewski's paper "Sheaves
of Banach spaces" to which Karl Heinrich Hofmann refers so admiringly in
his exquisite *Laudatio* in the present issue of these *Notices*.
Bernhard Banaschewski became an Editorial Board member of QM in 1987.

Banaschewski's visit of
July-September 1979 to UCT marked the beginning of his legendary and always
well attended series of lectures, in our Topology Seminar, on Frame Theory
(later to be styled "Pointfree Topology"). The lectures sometimes exceeded
three hours! It should not be inferred that we had more stamina or more
enthusiasm than we have now. It was merely the case in those simple and
uncluttered days that everybody (at least in a rather privileged university)
had more time. Let us further acknowledge what an astonishing range of
interconnected mathematical fields he handled in his research and in his
teachings. The SAMS Annual Congress held at UCT in October 1979 furnished
a nice sample. Banaschewski was at UNISA during that month, but he came
down to Cape Town for the Congress, where he gave an invited plenary talk
on "Banach spaces in other worlds". During the same Congress we ran
a series of special sessions on "Categorical Algebra and Topology", seventeen
talks in all. One of those talks was by Banaschewski: "When are the divisible
Abelian groups injective?" This title was intended to unsettle every law-abiding
algebraist for whom the well-known answer was "always". But in other "worlds"
the answer was different. The talk in effect was a strikingly transparent
sample of topos theory for outsiders. (It soon became a paper by Banaschewski
in QM **4**(1981), 285-307.)

As mentioned in the previous document, Banaschewski has been visiting the Topology Group (later called the Topology and Category Theory Group) at UCT every single year from 1979 right up to the present. Without attempting anything like a survey or progress report on the resulting work, I shall point out a few milestones such as theses and conferences along the years. Given the focus on the UCT-Banaschewski collaboration, some major areas of the work of the Topology Group necessarily lie outside the present purview. Thus the main contributions and outputs of several present and past members, and even some of their names, will not be told or listed here.

Regarding theses, it is
good to mention that every beginning research student in our group took
a number of courses appropriate to our and the student's interests. There
was often a course on pointfree topology by Banaschewski, in addition to
his seminar lectures. In this way a large proportion of our students were
attracted to his teachings, and they then had the benefit of his generous
involvement in their progress. (Even the MSc theses are worth telling about,
because in our system they are substantial efforts, examined by two international
referees.)

In 1979 my PhD student
Christopher Gilmour was working on a topological category of non-topological
spaces, the zero-set spaces, also known as Alexandroff spaces. Exposure
to Banaschewski's lectures on frame theory made Gilmour realise that (co-)zero-set
structures had a pointfree analogue, the sigma-frames. These are a loosening
of frames in which only countable joins are considered. The appropriate
duality was set up and the relations with uniformities and realcompactness
were studied. The PhD was awarded to Gilmour in 1981, and sigma-frames
continued to be an important aspect of the UCT-Banaschewski collaboration.

Asymmetry in general topology, comprising the study
of ordered spaces, bitopologies, quasi-uniformities and quasi-metrics,
had been one of our themes at UCT since 1968. In 1981 Banaschewski,
Hardie and I defined the pointfree analogue of a bitopological space, called
a biframe, and developed the corresponding theory as far as the Stone-Cech
compactification of biframes (even with a constructive proof leaning on
the proof of the corresponding result for locales by Banaschewski and Mulvey
from a 1980 paper of theirs). This was presented at the *Symposium on
Categorical Algebra and Topology* at UCT in June/July 1981, and our
paper, "Biframes and bispaces", appeared in the proceedings of the Symposium,
again a special issue of QM, volume **6**, parts 1-3, (1983), edited
by Hardie and myself.

The next milestone was the *Winter School on
Topos Theory* at Rhodes University, Grahamstown, July 1984, organised
by Keith Hardie and Gert Naudé. (How unreal it feels now to
recall that in 1984 the CSIR in Pretoria still had a National Research
Institute for the Mathematical Sciences, and that strong research in pure
as well as applied mathematics was done there. Naudé was a member
of that Institute, wrote papers and reports on applications of category
theory, and participated in our symposia from the beginning. Our proceedings
issues in QM **2 **(1977), QM **6
**(1983) and QM **9** (1986)
had papers by him. He died in 1993.) The five mornings of the Winter
School were devoted to courses of lectures on topos theory, given in rotation
by Bernhard Banaschewski and Chris Mulvey of Sussex. One spent the cold
afternoons trying to catch up. A photograph of the participants appears
below, including Gert Naudé and his wife Cornelia.

In July 1985 we had the *Conference on Classical
and Categorical Algebra* on the campus of the University of Natal²
at Durban. A serious attempt at interdisciplinary stimulation, it was organised
by the Topology Group in collaboration with the Mathematics Department
of UND. It attracted some leading international practitioners of both kinds,
and was deemed a success. The reader can look up the proceedings, which
comprised the entire Volume **9** (1986) of QM (again edited by Hardie
and me) and contained a partly topos-theoretic contribution by Banaschewski
and myself, "Thoughts on the Cantor-Bernstein Theorem".

Banaschewski's sixtieth birthday on 26 March 1986
was celebrated nearly a week long by an illustrious conference at McMaster,
with some of his South African friends participating. The proceedings,
edited by his friend and collaborator Evelyn Nelson, appeared after her
death in the *Journal of Pure and Applied Algebra*, **68 **(1-2),
(1990).

Also in 1986, John Frith was awarded a PhD at UCT
on his thesis "Structured frames", supervised by Keith Hardie. Frith showed
how to define and erect compatible quasi-uniform structures on frames by
using conjugate cover pairs. He developed the congruence lattice of a frame
as a viable alternative to constructing the assembly in terms of nuclei,
showing in particular that the congruence lattice appears naturally and
functorially as a normal, completely regular biframe, the functor being
the point-free analogue of the Skula functor from topological to bitopological
spaces. Subsequent results soon followed, e.g. in a paper by Banaschewski,
Frith and Gilmour in the *Pacific Journal of Mathematics* **130 **(1987),
209-213

Keith Hardie's sixtieth birthday in 1989 led to
a festschrift put together by the Topology Group with the support and encouragement
of Siegfried Grässer, then Managing Editor of QM. There were papers
by algebraists, categorists and topologists, including Keith's own PhD
(Cambridge, 1957) supervisor Peter Hilton. Banaschewski was represented
by a joint paper with me, "Strong zero-dimensionality of biframes and bispaces".
The *Festschrift* appeared in QM **13** (3-4), (1990).

Here a tribute to H.S.P. (Siegfried) Grässer
(1935-1990) is in place. He was Professor of Mathematics at UNISA, active
on the Council of SAMS for many years until his final illness, and one
of the driving forces behind QM since its inception. From the early 1970's
we at UCT had his enthusiastic cooperation whenever there was an international
visitor who could be shared between institutions. This was a good case
of synergy out of economic necessity. In this way Grässer became a
good friend of Herrlich, of Banaschewski, and of many of us. Siegfried's
network of international contacts was huge. He was devoted to QM, and would
have noted with pleasure how the exertions of Wesley Kotzé (Rhodes)
and Koos Grobler (Potchefstroom) have carried this journal of SAMS through
years of financial desperation until it now once more seems to be on an
even keel.

The decade 1989-1999 saw a spate of theses within
the Topology Group, of which I shall mention only those influenced by or
connected with Banaschewski. First, Joanne Walters received an MSc in 1989
under Gilmour's supervision on a thesis about uniformities on sigma-frames
and the cozero part of uniform frames. She proceeded to the Université
Catholique de Louvain to work under Francis Borceux. There she received
a Belgian Doctorate of Science, with Gilmour as external co-supervisor,
on the thesis "Completeness and nearly fine uniform frames", in 1995, under
her wedded surname Walters-Wayland. Meanwhile, Anneliese Schauerte had
completed an MSc on partially ordered spaces under my supervision in 1988
and gone to McMaster, where she received a PhD in 1992 under Banaschewski's
supervision on a thesis, entitled "Biframes", devoted to biframe properties
such as compactness, coherence, continuity and connectedness. She promptly
returned to take up a post at UCT. Next, in 1993, my student Kwena Rufus
Nailana received an MSc on strongly zero-dimensional partially ordered
spaces. In 1994 Vaclav Vajner, studying under Hubertus Bargenda, earned
a PhD on a rather purely categorical thesis about reflective hulls, which
however contained a remarkable application of the results by Frith, Banaschewski
and Gilmour [1986, 1987] on the congruence lattice of a frame. Namely,
Vajner showed that the category of complete Boolean algebras is not an
intersection of reflective subcategories of the category of frames, and
that the latter category is the strongly epireflective hull of the said
subcategory. Also in 1994, Gilmour's student Nizar Marcus received an MSc
for a contribution to realcompactifications and compactifications of frames,
using sigma-frame methods. In 1995 Angela Robertson, under Gilmour's supervision,
was awarded an MSc for "Normal bases and compactifications of frames".
In 1998 Marcus, still under Gilmour, received a PhD for the thesis "*E*-compactness
in point-free topology". Marcus is at the nearby University of the
Western Cape (UWC). Then, in 1999, Gilmour's student Phethiwe Matutu was
awarded a PhD for a thesis on bi-sigma-frames*, *the asymmetric version
of sigma-frames, in which she proved the categorical equivalence of the
compact regular bi-sigma-frames with the stably continuous sigma-frames.
Matutu teaches at the nearby University of Stellenbosch.

As if the last decade of the millennium was not
hectic enough, the Topology Group also organised the *Symposium on Categorical
Topology (SoCaT94). *It was held in November 1994, lasted a whole week,
and was dedicated by my kind colleagues to my sixtieth birthday. There
were about fifty participants from twelve countries. A charming feature
of the programme was the two courses of workshop lectures which started
off the mornings, after which the rest of each working day consisted of
talks, thirty-three in all, by other participants. The one course was by
Banaschewski, on "Completions of metric, uniform and nearness frames" ³.
The other was by Herrlich, on "Compactness and compactifications", describing
the role of choice principles in this theory. The refereed proceedings,
comprising 27 research papers but not the two workshop courses, came out
as follows:

** Festschrift...incorporating Proceedings of
the Symposium on Categorical Topology, University of Cape Town, 1994,**
edited by B. Banaschewski, C.R.A. Gilmour and H. Herrlich. Department of
Mathematics and Applied Mathematics, University of Cape Town, 1999, 271
pp. ISBN 0-7992-1962-2.

Banaschewski's seventieth birthday was honoured
at UCT in July 1996 with the *BB Fest 96*, which was also called a
"Conference on Category Theory and its Applications to Topology, Order
and Algebra". There were over 60 participants, of whom I cannot resist
naming the most senior: John R. Isbell, of SUNY at Buffalo. I can also
mention Bob Lowen, Editor-in-Chief of *Applied Categorical Structures,*
who encouraged us to submit the conference papers to that journal. The
resulting proceedings appeared in a twofold shape, as follows.

**Papers in Honour of Bernhard Banaschewski.
Applied
Categorical Structures, **Volume

The above collection of 23 papers was reprinted,
with 7 additional papers, as the following book.

**Papers in Honour of Bernhard Banaschewski.
***Proceedings
of the BB Fest 96, a Conference held at the University of Cape Town, 15-20
July 1996, on Category Theory and its Applications to Topology, Order and
Algebra.* Edited by Guillaume Brümmer and Christopher Gilmour.
Kluwer Academic Publishers, Dordrecht, September 2000, 504 pp.

Hardbound, ISBN 0-7923-6519-4. NLG 300.00/USD
147.00/GBP 93.00

The publisher's note on the back cover reads:

"Arising from the 1996 Cape Town conference in
honour of the mathematician Bernhard Banaschewski, this collection of 30
refereed papers represents current developments in category theory, topology,
topos theory, universal algebra, model theory and diverse ordered and algebraic
structures. Banaschewski's influence is reflected here, particularly in
the contributions to pointfree topology at the levels of nearness, uniformity
and asymmetry. The unifying theme of the volume is the application of categorical
methods. The contributing authors are: D. Baboolal, P. Bankston, R. Betti,
D. Bourn, P. Cherenack, D. Dikranjan/H.-P. Künzi, X. Dong/W. Tholen,
M. Erné, T.H. Fay, T.H. Fay/S.V. Joubert, D.N. Georgiou/B.K. Papadopoulos,
K.A. Hardie/K.H. Kamps/ R.W. Kieboom, H. Herrlich/A. Pultr, K.H. Hofmann,
S.S. Hong/Y.K. Kim, J. Isbell, R. Jayewardene/ O. Wyler, P. Johnstone,
R. Lowen/P. Wuyts, E. Lowen-Colebunders/C. Verbeeck, K.R. Nailana, J. Picado,
T. Plewe, J. Reinhold, G. Richter, H. Röhrl, S.-H. Sun, A. Tozzi/
V.Trnková, V. Valov/D. Vuma, S. Veldsman."

The story does not end here, because the collaboration
continues. For example, Banaschewski participated with his usual exuberance
in our *TopoSymCape 97* at UWC, and in the huge meeting of the *Pan-African
Congress of Mathematicians* at UWC in January 2000. - I see that
I have not yet mentioned the current leader of the UCT group, J.J.C. (Japie)
Vermeulen, known for his work in topos theory, locales and localic groups,
some of it joint with Banaschewski. However, as stated already, I did not
wish to survey, nor to report.

Apropos of toposes: Two new instalments from the
long-standing collaboration between Banaschewski and Mulvey have just appeared,
under the collective title "The spectral theory of commutative C*-algebras",
a total of 64 pages in QM **23**(4) (December 2000). For this work,
alas, UCT can claim no credit at all. But QM is sailing ahead!

Soon Bernhard's seventy-fifth birthday (26 March
2001) will be celebrated at the *Fourth Ibero-American Conference on
Topology and its Applications* in Coimbra (18-21 April 2001). Already
his next visit to UCT (September 2001-January 2002) has been planned. He
and his wife Angela have again booked the same fifth floor apartment, so
familiar to many friends, with the balcony looking out over the city centre
and Table Bay toward Signal Hill.

**Bernhard Banaschewski's many friends, admirers and colleagues in**

**South Africa hereby affectionately wish him and Angela a happy**

**celebration of his seventy-fifth birthday and a safe return!**

The above article was published, together with
ten photographs from the years 1982-2000, in the ** Notices of the South
African Mathematical Society, 32 **No.1 (April 2001), 27 - 38.

While that issue of the ** Notices** was
at the printers, an extremely sad event occurred: Our beloved UCT colleague
and collaborator, the noted worker on locales and topos theory,

¹Dis
die dyk en die tyd van die epiloog in: J.C. Kannemeyer, *Langenhoven*,
pp. 670-672. Tafelberg-Uitgewers,Kaapstad,
1995.

²The
choice of venue was not unrelated to the availability of a grant from the
Hanno Rund Research Fund, nor to the fact that Banaschewski and the distinguished
Rund had become friends in Canada in 1955, and that Rund in 1956-1957 had
tried to attract him to a post at UND, and had nearly succeeded.

³Published
as:B. Banaschewski, Completions
in pointfree topology, *Lecture Notes in Math. and Applied Math.*,
No. 2/96, Dept. of Math. and Applied Math., Univ. Cape Town, 1996.