NEW: Bob Paré and Category theory at Dalhousie
Our friend and colleague, Bob Paré, recently turned 80 years old. It therefore seems appropriate at this time to review Bob’s contributions to Category theory at Dalhousie, which has been and currently remains a world center for the field. A partial biography and list of his collaborators is included below.
After completing his PhD, in 1969, on “Absolute Colimits” at McGill with Lambek, Bob joined F.W. Lawvere’s Category Theory group at Dalhousie as a postdoctoral fellow. For a year during the two years that group existed, Chris Howlett, a PhD student at McMaster, was also part of the group. When Howlett returned to McMaster in 1972, Richard Wood was looking for a place and supervisor to do a PhD, and he recommended Bob as follows: “He understands hom functors! Richard – consider what this means!” That might sound like an odd sort of recommendation but within a few months Richard was sitting in Bob’s office while he showed him on his blackboard what was soon to be one of Bob’s most famous papers. He proved, for an elementary topos \mathbb{E}, with subobject classifier \Omega, the hom functor \mathbb{E}(-,\Omega):\mathbb{E}^{{\rm op}}\to\mathbb{E} is monadic. Amongst other things, Bob’s result showed that existence of finite colimits in an elementary topos followed from the other axioms. Bob called his paper “Colimits in Topoi”. A few weeks later, Peter Freyd contacted Bob to ask if he could have the paper to publish in JPAA.
That was an exciting year. Bob and Dietmar Schumacher from Acadia started the Category Theory Seminar at Dalhousie. Luzius Grunenfelder was a faculty attendee. There were several graduate students other than Richard including Vipin Sehgal, Ioana Schiopu and David Lever (before he went to Stony Brook for a year) who were part of the group, and the Seminar was also attended by several talented undergraduate students and some non category-theory graduate students. Peter Schotch of Philosophy also often showed up. In the years 1972-1973 there were many, many talks on “Indexed Categories”. The idea was that, for any object I in a topos \mathbb{E}, the Category \mathbb{E}/I, whose objects are arrows f:X\to I in \mathbb{E}, with codomain I, can serve as the category of I-indexed families of objects of \mathbb{E}. It was known that \mathbb{E}/I is also an elementary topos and that fact had already been dubbed the The Fundamental Theorem of Topos Theory. But the work of Bob and Schumacher contributed enormously to a deeper understanding of \mathbb{E}/I.
In 1973-74 Bob Rosebrugh became a graduate student of Bob and, soon after him, David Lever returned as a student and Javad Tavakoli joined too. Bob had a number of research visitors and postdoctoral fellows, and Dalhousie for the second time became a world center for Category Theory. In addition, many of Bob’s students found positions in the Atlantic region and soon there were researchers in cognate disciplines such as Hopf Algebras too. Grunenfelder was a Hopf algebraist, and he was joined by Margaret Beattie at Mount Allison and Mitja Mastnak at Saint Mary’s. Some years earlier, Robert Dawson, who had been a Dalhousie undergraduate, obtained his PhD at Cambridge and secured a position at Saint Mary’s. Bob’s influence had already been felt throughout the Atlantic region by the mid 1980s and many other researchers passed through.
Around 1988, motivated by his work with Michael Makkai on accessible categories, Bob developed an interest in double categories, a concept that had been introduced by Charles Ehresmann in 1963, and had both an algebraic and geometric appeal, but had not been studied in much detail for their own sake. (Double groupoids had been used in homotopy theory in the seventies by Ronnie Brown.) In the late 1980s graphical representations for composing/pasting 2-cells, so called pasting diagrams or pasting schemes, were introduced to category theory, and it would be natural to ask whether this type of representation could also work for double categories, especially since the double cells with their rectangular forms naturally gave rise to tilings. Bob had thought about it and realized that there was an obstruction in the form of what is now known as the pinwheel diagram. During an international category theory conference in Como in 1990, Bob told Robert about this and this led to the paper ”General associativity and general composition for double categories”, published in Cahiers in 1993.
Dorette Pronk, learned about their work through two presentations at the International Category Theory Conference in 1992 (her first international conference in category theory). Dorette was intrigued, but one of her fellow graduate students told her not to work on that, because it is going to be “way too popular”. Bob spent the next 30 years on showing the category theory community various aspects of the beauty of double categories, but there were not many ”early adopters”. He worked with Marco Grandis on a series of papers that can be viewed as foundational for the theory of double categories, developing the notions of adjunctions and limits. Susan Niefield, who had been a postdoctoral fellow in the department and continued to return as a regular visitor, worked on gluing constructions and tabulators, Bob developed many examples, coming mostly from algebra, and established various ways in which double categories can be used to better understand 2-categories, most notably showing that weighted limits can be better understood as double limits.
Dorette’s interest in Bob’s work, as well as his welcoming demeanor, led her as a new PhD to apply for a postdoctoral fellowship at Dalhousie. Bob was very interested in her work on orbifolds, represented by internal groupoids in topological spaces, so during her postdoctoral fellowship she did not work on double categories. However, when she returned as a faculty member, she developed an interest in using double categories to better understand localizations of categories. This led to a series of papers (with Bob and Robert) that used bicategories and double categories to freely add adjoints to the arrows in a category, and further explorations of the span construction. Related work was done by Michael Shulman on what he called framed bicategories and then on virtual double categories, also with Geoff Cruttwell, a former Dalhousie graduate student.
Double categories gained increased traction when John Baez and others at the Topos Institute started to use them successfully in applied category theory, specifically in networking theory and categorical logic. This was partially made possible by Bob’s work on the Yoneda theory for double categories. Bob also continued to inspire young category theorists at the International Category Theory conferences: he met with Raould Koudenburg, a new category theorist interested in double categories, after CT 2016 (which had been held in Nova Scotia), and inspired David Jaz Myers through conversations during CT2017 in Vancouver. More recently, Bob’s ideas on using certain double categories as double Lawvere theories have been picked up by students in Japan and in Halifax. Double categories and related work developed at Dalhousie in terms of equipments, have also been picked up by the infinity category theory community and play an important role in the development of a framework for formal category theory.
In 2022 the interest in research on double categories had grown so far that the first ”Virtual Double Category Workshop” was held, followed by a second one in 2024. There are now so many double category talks at the International CT meetings that there is no need to introduce the notion of double category anymore. The field has indeed finally become “popular”,
and this is in large part due to Bob’s contributions and encouragement of new graduate students.
Throughout his working years at Dalhousie Bob adhered to a ”vow of 2-dimensionality”, perhaps in part inspired by observations he and Robert made in their paper on what a free double category is like. Some of the nice features taken for granted in the graphical representations in two dimensions do not apply in three dimensions. Since Bob’s retirement he has worked on intercategories, a 3 dimensional generalization of double categories (which are triple categories that satisfy a number of constraints). He continues to attend seminars at Dalhousie and interact with the algebra group.
More recently Peter Selinger, and thereafter his graduate student Julien Ross, joined Dalhousie faculty, partially overlapping with Bob. Peter and Julien work on programming languages for quantum computing and quantum circuit theory. Some of this work involves type systems that can be modeled in indexed categories, which is a field in which Bob has made crucial contributions. Continuing the tradition established by Bob, there are currently students and postdoctoral fellows working on type theory (other than for quantum computing) and on the theory of double categories, another area that was pioneered by Bob. Theo Johnson-Freyd, who recently joined Dalhousie, works on higher categorical algebra, especially as it relates to
mathematical physics. Higher categories is another area that Bob has contributed to.
In summary, throughout Bob’s career he has made an enormous contribution to category theory at Dalhousie, not only through his own work but by also creating a flourishing research culture through the beauty he uncovers and through the questions he asks. It should also be said that Bob’s contributions to the Department are very much appreciated by all of his colleagues and friends.
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Alan Coley, Robert Dawson, Dorette Pronk, Peter Selinger and Richard Wood
