bio  website  bangor.ac.uk/r.brown 

location  
age  80  
visits  member for  5 years, 2 months 
seen  10 hours ago  
stats  profile views  4,531 
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.
7h

awarded  Nice Answer 
Jul 29 
revised 
Why are we interested in the Fundamental Groupoid of a Space?
slight change of emphasis 
Jul 22 
comment 
Exact sequences in homotopy categories
I have put a talk entitled "A philosophy of modelling and computing homotopy types" given to CT2015, in Aveiro, on my preprint page pages.bangor.ac.uk/~mas010/brownpr.html . This shows some explicit colimit calculations, which are more precise than exact sequences. 
Jul 9 
revised 
Pullback of a fibration along a homotopy equivalence and homotopy classes of sections
full explanation added 
Jul 5 
comment 
Higher refinement of Seifertvan Kampen theorem on the language of hocolim
I get confused by all these ideas since the fundamental groupoid $\pi_1(X)$ of a space $X$ is a strict groupoid, and the version of the Seifertvan Kampen Theorem useful for computation involves $\pi_1(X,C)$, the fundamental groupoid on a set of $C$ of base points. The so called "higher groupoids" of various writers are not groupoids at all, but lax versions, so the analogy is not clear. Another source of confusion is that homotopy groups are defined only for spaces with a given base point. 
Jun 24 
revised 
Higher refinement of Seifertvan Kampen theorem on the language of hocolim
deleted repetition 
Jun 24 
comment 
Higher refinement of Seifertvan Kampen theorem on the language of hocolim
Lurie's version is also discussed at mathoverflow.net/questions/102295/… My answer below explains more on my ideas of such higher Seifertvan Kampen theorems. 
Jun 24 
revised 
Higher refinement of Seifertvan Kampen theorem on the language of hocolim
added an abstract of the Aveiro presentation 
Jun 20 
revised 
Higher refinement of Seifertvan Kampen theorem on the language of hocolim
extra explanation of history 
Jun 20 
answered  Higher refinement of Seifertvan Kampen theorem on the language of hocolim 
Jun 10 
revised 
Compelling evidence that two basepoints are better than one
added a link 
Jun 8 
comment 
What is the intuition of connections for cubical sets?
Vezzani (arXiv:1405.4508) and cited papers there for relevance to motivic theory. In this area the corresponding simplicial methods have disadvantages! See also 116. (with F.A. ALAGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71118. 
Jun 8 
comment 
What is the intuition of connections for cubical sets?
I'll add a couple of later references. Homology Homotopy Appl. Volume 14, Number 1 (2012), 133158. Cubical approach to derived functors Irakli Patchkoria 
Jun 7 
comment 
Pullback of a fibration along a homotopy equivalence and homotopy classes of sections
You can contact me by email if you want. One seems to need the old trick: if abc is defined in a category, and ab, bc are isomorphisms, then a,b,c, are iso also. 
Jun 6 
revised 
Pullback of a fibration along a homotopy equivalence and homotopy classes of sections
added a comment on further studies of maps 
Jun 6 
revised 
Pullback of a fibration along a homotopy equivalence and homotopy classes of sections
added a reference 
Jun 6 
answered  Pullback of a fibration along a homotopy equivalence and homotopy classes of sections 
May 30 
comment 
Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?
@Andy Putnam The question asked in the last paragraph of the main question seems to be about the whole structure of the usual homology theory, starting with singular theory, and continuing with computation via cellular chains. One also has to look at the whole history of algebraic topology, as glimpsed in the book and presentation cited by me. These show a direct and homotopically defined way to the cellular complex, but in a more powerful form with operators, and using crucially JHC Whitehead's notion of crossed module, a term not in Hatcher's book. 
May 24 
awarded  Good Answer 
May 22 
comment 
What's a groupoid? What's a good example of a groupoid?
This discussion should be linked to mathoverflow.net/questions/40945/… 