bio  website  bangor.ac.uk/r.brown 

location  
age  79  
visits  member for  4 years, 6 months 
seen  31 mins ago  
stats  profile views  3,889 
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 (197173), Philip Higgins (19742005), and JeanLouis Loday (19811985), 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 coauthors P.J. Higgins and R. Sivera.
2d

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. 
2d

comment 
When is a topological space the homotopy colimit of an open covering?
My attitude to Seifervan Kampen thoerems is that they are about direct computation of a strict homotopical invariant as an exact colimit. 
2d

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 Seifertvan 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 dun 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 
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 