6,669 reputation
2438
bio website bangor.ac.uk/r.brown
location
age 79
visits member for 4 years, 3 months
seen 1 hour 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.


Aug
19
revised Do Disjoint Unions and Fiber Products Commute?
gave a link to a PhD thesis
Aug
5
revised Generalized Categories for “Higher Homotopy Groupoids”
additional remarks, dated
Jul
22
revised Why are we interested in the Fundamental Groupoid of a Space?
added a remark on exact sequences for fibrations of groupoids
Jul
22
comment Compelling evidence that two basepoints are better than one
@HJRW: Just to show the value of a more general viewpoint, groupoids and higher van Kampen theorems led Loday and I to the notion of nonabelian tensor product of groups which act on each other; the current bibliography on my web page has 131 items, mainly from group theorists (only 5 with my name on). See also answers to mathoverflow.net/questions/175923.
Jul
17
awarded  Nice Answer
Jul
17
revised Why are we interested in the Fundamental Groupoid of a Space?
Three more significant uses of the fundamental groupoid
Jul
14
revised Why are we interested in the Fundamental Groupoid of a Space?
typo
Jul
14
revised Why are we interested in the Fundamental Groupoid of a Space?
added an extra point about the proof
Jul
13
answered Why are we interested in the Fundamental Groupoid of a Space?
Jul
13
awarded  Nice Answer
Jul
11
answered How does one justify funding for mathematics research?
Jul
9
revised Generalized Categories for “Higher Homotopy Groupoids”
added a link to stackexchange
Jul
8
revised Generalized Categories for “Higher Homotopy Groupoids”
added an explicit reference
Jul
7
revised Generalized Categories for “Higher Homotopy Groupoids”
addeda new link
Jul
6
awarded  Nice Answer
Jun
30
comment Grothendieck's Homotopy Hypothesis - Applications and Generalizations
I don not know details of the Joyal-Tierney model of $3$-types, but crossed squares as models of $3$-types were published by Loday in 1982, and were used in a paper of mine and Nick Gilbert, published in 1988, and in which the model was related to automorphisms of crossed modules. Advantages of crossed squares are: they are a strict model; are values of a homotopically defined functor; colimits are commonly directly related to geometry and algebra; they extend in a natural way to all dimensions; the concept applies to other algebraic structures than groups, e.g. Lie algebras.
Jun
30
comment Grothendieck's Homotopy Hypothesis - Applications and Generalizations
I agree with Todd $(\infty,1)$-groupoids are not really algebraic, and neither are $k$-invariants. As models of $2$-types, crossed modules (over groupoids) seem pretty good, as their limits and colimits, where the latter correspond to many gluings of $2$-types, are good for examples and calculations. These gluings are also related to the convenient compositions which you get in cubical theories. These (partial) compositions are often neglected, but seem to me more "algebraic" than the Kan condition.
Jun
29
answered What's special about the Simplex category?
Jun
23
revised Computing homotopies
added a comment
Jun
22
comment Grothendieck's Homotopy Hypothesis - Applications and Generalizations
Maybe "uncomfortable" was the wrong word, but the idea of using structured spaces, or say "filtered topoi", has not taken off! There is a methodological case that is one is "testing" a space with a structured object such as $I^n$ or $\Delta^n$ then one should use relevant/related structure on the space, to allow more scope for the tool one is using.