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bio website bangor.ac.uk/r.brown
location
age 80
visits member for 5 years
seen 5 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.


2d
comment What's a groupoid? What's a good example of a groupoid?
This discussion should be linked to mathoverflow.net/questions/40945/…
May
20
revised Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?
added a link
May
19
awarded  Yearling
May
14
answered Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?
May
14
awarded  Necromancer
May
10
awarded  Nice Answer
May
2
answered Exact sequences of pointed sets - two definitions
Mar
30
revised Has any attempt been made to classify finite groupoids?
typo and word order
Mar
30
comment Has any attempt been made to classify finite groupoids?
@Qiaochu Yuan: Groupoids equipped with a $G$-action where $G$ is a group occur in the study of orbit spaces, where the groupoid is the fundamental groupoid $\pi_1(X)$. See Chapter 11 of Topology and Groupoids, where the "orbit groupoid" is related to the fundamental groupoid of the orbit space!
Mar
29
revised Has any attempt been made to classify finite groupoids?
typo
Mar
29
awarded  Necromancer
Mar
27
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)