6,236 reputation
2234
bio website bangor.ac.uk/r.brown
location
age 79
visits member for 3 years, 11 months
seen 17 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.


Apr
8
comment Fundamental group of a manifold with an $S^1$-action
Here is a follow up question. Chapter 11 of "Topology and Groupoids" deals with orbit spaces and actions of groups on groupoids, obtaining a theorem that for a discrete, group $G$ acting discontinuously on a Hausdorff space $X$ with some good local properties, the fundamental groupoid of the orbit space $X/G$ is isomorphic to the orbit groupoid of the fundamental groupoid $\pi_1 X$ by the action of $G$. This result includes work of A. Armstrong. Question: Are there possible extensions to actions of topological groups, such as $S^1$?
Apr
8
revised Compelling evidence that two basepoints are better than one
displayed a question
Apr
1
answered Extending binary operation used by homotopy classes
Mar
31
revised Compelling evidence that two basepoints are better than one
added a picture
Mar
30
comment Transporting algebraic structure along adjoint equivalences
Is this reference, arXiv:1403.3254, on fibrations of ordered groupoids, relevant to your question?
Mar
30
revised Functors and coverings
gave an additional linked reference .
Mar
23
comment Pushout of categories along embeddings gives homotopy pushout?
Do all these "evil" connotations apply to groupoids, and higher groupoids? I like colimits of groupoids and higher groupoids, and have used them for specific homotopy invariant calculations. The higher case led to the notion of nonabelian tensor product of groups, in work with Loday. Can someone please give me an example of an algebraic calculation using homotopy 2-colimits?
Mar
14
awarded  Necromancer
Mar
10
revised Postnikov's algebraic reconstruction of cohomology from homotopy invariants
typo
Mar
9
revised Functors and coverings
added a comment about actions
Mar
9
revised Postnikov's algebraic reconstruction of cohomology from homotopy invariants
clarified some points and an additional reference
Mar
9
revised Postnikov's algebraic reconstruction of cohomology from homotopy invariants
clarified some points and an additional reference
Mar
9
answered Postnikov's algebraic reconstruction of cohomology from homotopy invariants
Mar
8
answered Functors and coverings
Feb
25
comment Geometric interpretation of the fundamental groupoid
The action of $G$ on the fundamental groupoid leads to a good result that in useful conditions, the fundamental groupoid of the orbit space is the "orbit groupoid" of the fundamental grouoid: see my book "Topology and Groupoids", Chapter 11.
Jan
29
comment Compelling evidence that two basepoints are better than one
@Ryan: "terminology preferences" can be very useful. An extreme example is Roman numerals versus Arabic numerals. I also tend to take the line expressed by the question: "Is there compelling evidence that that one can do better without one hand strapped behind ones back?" Or, as Gian-Carl Rota wrote on another matter: "Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures.” The idea of groupoids has led me to new worlds, and modes of expression. What fun!
Jan
25
comment Is there an analog of Sperner's lemma for the Hopf invariant?
This comment is algebraic rather than "combinatorial" so I don't give it as an answer. But Loday and I in Topology 26 (1987) 311-334, defined a tensor product $G \otimes H$ of groups which act on each other and on themselves by conjugation, and showed that $\pi_3SK(G,1)$ is the kernel of the "commutator map" $\kappa: G \otimes G \to G$. When you take $G=\mathbb Z$, you get the Hopf map is $1 \otimes 1$.
Jan
23
comment Homotopy category of groupoids
Why does one want to move from $Gpd$ to $HO(Gpd)$?
Jan
21
comment What are your favorite concrete examples of limits or colimits that you would compute during lunch?
@Todd: I mention that the groupoid example of the integers was in essence in my 1967 paper on van Kampen's theorem and was in the 1968 book on Topology.
Jan
20
comment What are your favorite concrete examples of limits or colimits that you would compute during lunch?
@Todd: That example is a good one, and can also be phrased as a pushout in the category $Cat$ of small categories, noting that $Cat \to Sets$ is a bifibration of categories. Thanks also for the references.