7,454 reputation
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bio website bangor.ac.uk/r.brown
location
age 80
visits member for 5 years, 2 months
seen 10 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.


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 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.
Jun
6
revised Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
added a comment on further studies of maps
Jun
6
revised Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
added a reference
Jun
6
answered Pull-back 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/…