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bio website bangor.ac.uk/r.brown
location
age 80
visits member for 4 years, 10 months
seen 2 hours ago
I was a research student of Henry Whitehead till 1960, when he died suddenly at Princeton; then Michael Barratt, as new supervisor, suggested the problem of the homotopy type of function spaces $X^Y$, by induction on the Postnikov system of $X$. A particular problem was calculating $k^Y$ where $k$ is a cohomology operation. This worked out well. In doing this, I came across lots of exponential laws, essentially monoidal closed categories, and this led me to the notion of convenient category for topology, and my first two papers.

As a result of giving courses at BSc and MSc level at Liverpool, I got involved in writing a text published as "Elements of Modern Topology" with McGraw Hill in 1968, and the 3rd edition is now available as "Topology and Groupoids" from amazon, see my web page. It was writing this book, and trying to clarify certain points, such as the fundamental group of the circle, that got mew into the area of groupoids; this suggested the area of higher groupoids; research on this got going in the 1970s, and has been a major area in my work, with fortunate collaborations with Chris Spencer (1971-73), Philip Higgins (1974-2005), and Jean-Louis Loday (1981-1985), and excellent contributions from research students.

Since 2001 I was engaged in writing a Tract giving an exposition in one place of my work since 1974 with Philip Higgins and a number of others on (strict) higher homotopy groupoids and their applications. A first step was to make available my out of print book on Topology, and this was made available in 2006 under the title "Topology and Groupoids" see my web page. The higher dimensional work was published in August, 2011, as a European Mathematical Society Tract, Vol 15, under the title "Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids" with co-authors P.J. Higgins and R. Sivera.


7h
revised Has any attempt been made to classify finite groupoids?
typo
10h
awarded  Necromancer
2d
revised Has any attempt been made to classify finite groupoids?
added extra comments
Mar
25
revised What's special about the Simplex category?
added some extra references
Mar
24
revised What's special about the Simplex category?
more info on motives and cubical sets
Mar
16
comment Brandt's definition of groupoids (1926)
One of the fascinations of mathematics is how structures can be appropriate to help us understand the small and the large. A good example is monoidal categories which crop up for describing the family of braid groups and also of chain complexes. Charles Ehresmann wrote that his aim was to understand the structure of everything; that also includes the structure of structures!
Mar
14
comment Brandt's definition of groupoids (1926)
Martin writes "groupoids look very much like groups except that the product is only defined partially.". My definition of "higher dimensional algebra" is "the study of algebraic structures with partially defined operations whose domains are determined by geometric conditions", so that we can combine algebra and geometry, opening new worlds. Philip Higgins wrote the first paper on partial algebraic structures, "Algebras with a scheme of operators", Math. Nachr. 27 (1963) 115--132.
Mar
14
comment Brandt's definition of groupoids (1926)
It is not unusual for people to be influenced by a work and then forget about that as they write and rewrite. I have probably done this.
Mar
14
comment Brandt's definition of groupoids (1926)
I have no explanation. The paper listed under General Papers no 8 at people.math.ethz.ch/~knus/publications09.html states that it is believed that the groupoid axioms influenced the work of Eilenberg and Mac Lane.
Mar
14
revised Brandt's definition of groupoids (1926)
added a reference to a translation of the Reidemeister book
Mar
13
answered Brandt's definition of groupoids (1926)
Mar
12
comment Dimension leaking in homology as opposed to homotopy
@AlexDegty: The groupoid version of the SvKT using a set of base points which I published in 1967 gets over the path-connected assumption - see the presentation referred to in my answer, and mathoverflow.net/questions/40945/…
Mar
12
revised Dimension leaking in homology as opposed to homotopy
slight clarification
Mar
12
answered Dimension leaking in homology as opposed to homotopy
Mar
11
revised Compelling evidence that two basepoints are better than one
grammatical improvement to last sentence
Mar
11
awarded  Necromancer
Mar
11
revised Compelling evidence that two basepoints are better than one
added another link
Mar
7
answered Topological Grothendieck Construction
Mar
1
awarded  Necromancer
Feb
25
revised Do we still need model categories?
added a link to a recent talk