### My interest in monoidal bicategories

**An old article first posted in 1997**

**My interest in monoidal bicategories.**

**RFC Walters 7th April 1997**

**1966-70**In my PhD thesis (ANU) about category theory and universal algebra I used the 2-category structure of Cat to define things and hence became interested in pasting cells.

**1970**I visited Mac Lane (Chicago) as research associate and discovered that the category PA of presheaves on a small category is 'total' which lead to a collaboration with Ross Street beginning in 1970 of the 2-categorical properties of yoneda yA:A->PA. We were trying to characterize Sets in Cat but in analogy to studying 2 in Sets by looking at arrows A->PB (here PB means subsets of B) we studied functors A->PB.

**1974**I spoke on his work at a conference at UNSW organized when Peter Freyd was here by Max Kelly. The paper appeared much later in "Yoneda structures". The beginning of the study of Mod and Rel was already there.

**1980**I visited Milan and discovered that the sheaf condition can be expressed in terms of Cauchy completeness of categories enriched over a bicategory of 'relations' - this work appeared in Cahiers. A group of us studied categories enriched over bicategories, and bimodules between them. We looked at properties which lifted from the base bicategory to the bicategory of bimodules.

**1982**Lifting the tensor product lead to my idea with Lawvere and Carboni that the classical treatment of this in terms of symmetry could be explained in terms of a tensored one-object bicategory and a Eckman-Hilton argument - I gave a lecture on this on 26th January 1983 which inspired Ross to discover braided monoidal categories with Joyal (whose motivation was different).

**1983**I began the study with Carboni of Rel as a monoidal bicategory in Milan December 1983. It was completed in January 1985. Aurelio had previously done work in the direction of Freyd on relations regarding Rel as a category and using the mysterious 'modular law'.

**1985**I believe I lectured on Rel at Sussex. The paper had been submitted some time before and Scedrov had written a report recommending rejection, but it was eventually accepted. A third strand was that Mike Johnson and I had begun work (ask Mike when he began as my student) on an alternative version of the n-category generated by an n-simplex - the higher associative laws. (I had suggested to him originally a different problem: describing cech cocyles as categories enriched over an n-category.)

**1985**Mike and I presented this at Bangor. At around that point I began to be interested in computer science. I gave a couple of lectures in the Sydney Category Seminar. I occasionally tried to get divert Mike's attention in this direction. That took me off into distributive, extensive categories, only to return to monoidal bicategories (now with feedback) in 1994.

**1992**I began the study of concurrency with Nicoletta Sabadini.

**1994**After a visit of Bloom, seeing his work on iteration theories I gave a lecture at the Sydney Half-hour Seminar on machines as arrows (29th April). Since then Giulio Katis, Nicoletta Sabadini and I have worked extensively on this idea, one result being the 1997 paper "Bicategories of Processes". An earlier version of the paper was written presented at CATS '94. The work with Henry Weld on circuits, and the earlier work with Wafaa Khalil on 'functional processes' have been very influential - in fact there is an exercise in the 1992 Cambridge version on my book which appropriately interpreted is about the monoidal bicategory of Elgot automata.

Labels: category theory