6,988 reputation
2540
bio website bangor.ac.uk/r.brown
location
age 79
visits member for 4 years, 7 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.


1h
comment Compelling evidence that two basepoints are better than one
@HJRW: In mathematics, I feel there can often be a world of difference between "implicit" and "explicit", especially if the latter is followed by exploitation. See for example this paper PAUL LUNAU, Int. J. Algebra Comput., 15, 129 (2005). For me, the chief advantage of groupoids was that they led to higher homotopy groupoids, see presentations in 2014 at Paris, IHP, and Galway, on my preprint page.
Dec
12
comment What is operator tmf?
To throw out a possibly wild idea, this question might be related to this one: mathoverflow.net/questions/86617/…
Dec
8
comment Compelling evidence that two basepoints are better than one
@Ryan: You comment that "you have a covering space, so study that"; but which one? A universal cover of $X$ is defined by $X$ and a base point $x$. It is in fact the star, or costar, of $\pi_1 X$ at a base point $x$. See my preprint page for a recent talk at Galway which starts with 5 anomalies in algebraic topology, most of which are resolved using groupoids and/or cubes in some way.
Nov
24
comment When is a topological space the homotopy colimit of an open covering?
The invariants I have dealt with are of: spaces with many base points; filtered spaces; and $n$-cubes of spaces. The first two are dealt with in the EMS Tract vol 15 (2011) on "Nonabelian algebraic topology:...", and were published first in 1967 and 1981, respectively.
Nov
24
comment When is a topological space the homotopy colimit of an open covering?
My attitude to Seifer-van Kampen thoerems is that they are about direct computation of a strict homotopical invariant as an exact colimit.
Nov
23
comment When is a topological space the homotopy colimit of an open covering?
My question mathoverflow.net/questions/102295/… is relevant to your comment. Note that this sometimes called "small simplex" theorem has other proofs in the literature, and has not led to proofs of higher Seifert-van Kampen Theorems, to my knowledge.
Nov
23
answered When is a topological space the homotopy colimit of an open covering?
Nov
3
revised Why higher category theory?
moe explanantion
Nov
3
answered Why higher category theory?
Oct
13
comment General gluing theorem for adjunction spaces
I think the general result in a cofibration category is of interest but I would also like to know of significant topological examples where the closed hypothesis is dropped. If anyone wants to see the original proof it is available in this upload of Chapter 7 of Topology and Groupoids: pages.bangor.ac.ul/~mas010/pdffiles/TandGCh7.pdf
Oct
2
revised Compelling evidence that two basepoints are better than one
gave a recent link
Sep
28
awarded  Nice Answer
Sep
24
awarded  Autobiographer
Sep
12
awarded  Necromancer
Sep
9
revised Naturality of a Kunneth formula for cohomology
typos
Sep
9
answered Naturality of a Kunneth formula for cohomology
Sep
8
comment Decomposition vs filtration vs stratification
You should be interested in the comments of Grothendieck in section 5 of his "Esquisse d-un Programme" (1984) (matematicas.unex.es/~navarro/res/esquisseeng.pdf) which criticises the notion of topological space as inadequate for geometry, and discusses stratifications. See my preprint page pages.bangor.ac.uk/~mas010/brownpr.html for slides of a talk in June in Paris discussing the use of filtrations to define a strict cubical higher homotopy groupoid.
Sep
1
revised Is there an accepted definition of $(\infty,\infty)$ category?
gave an additional link, and one typo corrected
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