7,492 reputation
3243
bio website bangor.ac.uk/r.brown
location
age 80
visits member for 5 years, 3 months
seen 30 mins 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
18
awarded  Necromancer
Aug
17
comment Compelling evidence that two basepoints are better than one
@Ryan I just mention that looking at groupoids led me to higher homotopy Seifert-van Kampen Theorems and that led eventually to nonabelian tensor products of groups of which the bibiliography on my web page has 131 items, the majority by group theorists: see pages.bangor.ac.uk/~mas010/nonabtens.html.
Aug
17
answered Connection between Stalling's end theorem and Seifert-van Kampen Theorem
Aug
13
revised Why higher category theory?
added another mathoverflow link
Aug
8
comment Two graph structures on $\text{Hom}(G,H)$
Structures on $Hom(G,H)$ are discussed in this paper R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, `Graphs of Morphisms of Graphs', Electronic Journal of Combinatorics, A1 of Volume 15(1), 2008. 1-28. pdf available at pages.bangor.ac.uk/~mas010/pdffiles/graphmorphisms-v15i1a1.pdf . This also discusses directed and undirected graphs.
Aug
4
revised Compelling evidence that two basepoints are better than one
added another link
Aug
1
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 Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
full explanation added
Jul
5
comment Higher refinement of Seifert-van 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 Seifert-van 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 Seifert-van Kampen theorem on the language of hocolim
deleted repetition
Jun
24
comment Higher refinement of Seifert-van 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 Seifert-van Kampen theorems.
Jun
24
revised Higher refinement of Seifert-van Kampen theorem on the language of hocolim
added an abstract of the Aveiro presentation
Jun
20
revised Higher refinement of Seifert-van Kampen theorem on the language of hocolim
extra explanation of history
Jun
20
answered Higher refinement of Seifert-van 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. AL-AGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71-118.
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), 133-158. Cubical approach to derived functors Irakli Patchkoria
Jun
7
comment Pull-back 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.