User ronnie brown - MathOverflow

Answer by Ronnie Brown for Connected groupoids and action groupoids

2013-04-24T14:05:21Z

Here is my stackexcnage answer.

Another way of looking at this is to use the equivalence of categories between covering morphisms of a groupoid $P$ and actions of $P$ on sets. (Recall that a covering morphism $p:G \to P$ is a groupoid morphism having unique path lifting. Not necessarily unique path lifting gives a fibration of groupoids.) Given an operation of $P$ on a set $X$ then the corresponding covering morphism may be written $P \ltimes X$, an action groupoid, and thought of as a semidirect product because it is a special case of the semidirect product for an action of a groupoid $P$ on a groupoid $H$. For this one needs a morphism of groupoids $\omega: H \to Ob(P)$, where the latter is thought of as a discrete groupoid, and an element $w: x \to y$ in $P$ gives a morphism of groupoids $w_*: \omega^{-1}(x) \to \omega^{-1}(y)$. One has to be precise on conventions to get all this right, which I won't do here.

So a groupoid $G$ has a representation as an action groupoid whenever you are given a covering morphism $ G \to P$. This is closely related to Omar's answer, of course.

I'll add that more details of these ideas are in my book Topology and Groupoids.

Addition: May 19, 2013 Here is a version of Sam's nice argument but in the language of covering morphisms.

If $G$ is a group then its universal cover $p: T \to G$ is a covering morphism of groupoids such that $T$ is connected and has trivial vertex groups; this is determined by the action of $G$ on itself by left multiplication. The set of objects of $T$ is bijective with the set $G$ and $T$ is the indiscrete groupoid (also called tree groupoid) on its set of objects. (Note that if $S$ is a generating set for $G$ then $p^{-1}(S)$ is a graph, namely the Cayley graph of $(G,S)$.)

Now let $A$ be a connected groupoid with $X$ as its set of objects. Let $T$ be the indiscrete groupoid on $X$. Then for any object $x$ of $A$, $A$ is isomorphic to $A(x) \times T$. But if $G$ is as above, then $$1 \times p: A(x) \times T \to A(x) \times G$$ is a covering morphism.

The next question is whether this argument can illuminate the case $A$ is a $\Gamma$-groupoid.

Answer by Ronnie Brown for Groupoid actions on spaces

2013-05-18T21:00:38Z

Actions of a Lie groupoid are defined on p. 34 of K.C.H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids" LMS Lecture Notes Series no 213, 2005.

Note that in general for a groupoid action of $G$ on sets it is often convenient to follow C. Ehresmann and to insist on having a function $f: E \to Ob(G)$ so that $g \in G(x,y)$ maps the fibre of $f$ over $x$ to the fibre over $y$. In fact a standard equivalence is between actions on sets in this sense; functors $G \to Sets$; and covering morphisms of the groupoid $G$. But the point of the first definition is that this easily transcribes to the case $E$ is a topological space, as in Mackenzie's book. A full exposition of covering space theory based on covering morphisms of grouypoids, rather than actions, is given in the book now called "Topology and Groupoids", and was in the 1968 edition.

John Klein is also right to emphasise the covering space example. This leads to the idea that for the cellular homology of the universal cover of a CW-complex you actually need chain complexes with a groupoid of operators, rather than the usual group of operators. This idea was developed in a paper with Higgins (Proc Camb. Phil. Soc. (1990)) and is explained in the book Nonabelian algebraic topology, see for example Section 8.4.

Edit May 19: A simple and basic example of a groupoid acting on spaces generalises the case of a topological group acting on itself by left multiplication. A groupoid $G$ acts on the families of stars $St_G(x), x \in Ob(G)$, where $St_G(x)$ is the union of the sets $G(x,y)$ for all $y \in Ob(G)$, using the convention that if $g: z \to x, h:x \to y$ then $gh: z \to y$. If $G$ is a topological groupoid, then we get an action of $G$ on topological spaces.

An example of this is the case the space $X$ admits a universal cover. Then the fundamental groupoid $\pi_1 X$ may be topologised making it a topological groupoid. (R. Brown and G. Danesh-Naruie, ``The fundamental groupoid as a topological groupoid'', Proc. Edinburgh Math. Soc. 19 (1975) 237-244.) The star of $\pi_1 X$ at $x$ is of course the universal cover of $X$ based at $x$. (See also "Topology and Groupoids" 10.5.8, which deals with the case $(\pi_1 X)/N$ for $N$ a totally disconnected normal subgroupoid of $\pi_1 X$. )

Answer by Ronnie Brown for Why is Set, and not Rel, so ubiquitous in mathematics?

2013-05-18T21:17:08Z

I think there is a subquestion, or related (!) question, namely why is there little functional analysis of partial functions? Teaching real analysis to students makes it clear that it is all about partial functions $\mathbb R \to \mathbb R$ , each of which has a domain and range, which we often want to know. Now the solutions of differential equations with a parameter are often regarded as smooth in the parameter, but usually only with fixed domain. For example it is reasonable to suggest that the family of partial functions $f_y: x \mapsto \log(x+y)$ varies continuously with $y$, but of course the domain varies with $y$, so the answer is not so clear, especially if you suggest, why not?, that $f_y$ is, or should be, a smooth function of $y$.

Actually a web search on "partial functions" gives quite lot of hits, but I am not sure of anything definitive.

Answer by Ronnie Brown for Magic trick based on deep mathematics

2013-05-12T21:28:52Z

The coffee mug trick is also called the Philipine Wine Trick and should be related to the Dirac String Trick, which you can find by a web search, for example here and also in my presentation Out of Line, where rotations in 3-space are related to the Projective Plane.

A knot trick, I am not sure you would call it magic, has been shown to children and academics in many places. It rquires a pentoil knot of width 20" made of copper tubing, about 7mm diameter (made by a university workshop) shown in the following diagram:

It also needs some nice flexible boating rope. The rope is wrapped round the $x,y$ pieces according to the rule $$R=xyxyxy^{-1}x^{-1}y^{-1}x^{-1}y^{-1} $$ and the ends tied together, as in the following picture:

A member of the sudience is then invited to come up and manipulate the loop of rope off the knot, starting by turning it upside down. This justifies the rule $R=1$. Of course the rule is the relation for the fundamental group of complement of the pentoil, which can, for the right audience, be deduced from the relations at each crossing given by the diagram

and can be easily demonstrated with the knot and rope.

It is also of interest to have a copper trefoil around to compare the relations. One warning: the use of rope does not really model the fundamental group, so be careful with a demo for the figure eight knot!

I did the demo for one teenager and he said:"Where did you get that formula?" This demo knot has been well travelled, for many different types of audience; on one occasion the airline lost my luggage with the rope and I had to ask the taxi from the airport to to stop at a hardware store for me to buy d=some clothesline. I devised this trick for an undergraduate course in knot theory in the late 1970s.

Answer by Ronnie Brown for Compact open topology

2013-05-10T21:30:39Z

I have a weakness for pictures. So here is one

The above shows a function $f: \mathbb R \to \mathbb R$, a compact set $C$, and an open set $U$. The condition $f(C) \subseteq U$ is that the graph of $f$ passes through the shaded part shown in the picture.

Answer by Ronnie Brown for Homotopy equivalences preserving structure

2013-05-09T10:33:22Z

I think an answer is in

tom Dieck, Tammo Partitions of unity in homotopy theory. Composito Math. 23 (1971), 159–167.

With regard to the result on pairs given by Steve, it could be useful to note that the book Topology and Groupoids gives a result in 7.4.2(Addendum) which gives control over the homotopies involved. The utility of this is that it gives a key to one proof of a gluing theorem for homotopy equivalences, which was first given in the 1968 edition of this book and is applied by tom Dieck in his paper. The Addendum is as follows:

We are dealing with the situation $f:(X,X^0) \to (Y,Y^0)$ where each pair has the HEP.

Let $g^{0} :Y_{0} \to X_{0}$ be any homotopy inverse of $f^{0}$ and let $ H^0: f^0g^0 \simeq 1, K^0: g^0f^0 \simeq 1$ be homotopies. Then $g^0$ extends to a homotopy inverse $g$ of $f$ such that the homotopy $fg \simeq 1 $ extends $H^0$ while the homotopy $gf \simeq 1$ extends the sum $$ K^{0} + g^{0}H^{0}f^{0} - g^{0}f^{0}K^{0} $$ of the homotopies $$ g^{0}f^{0} = g^{0}f^{0}1_{X_{0}} \simeq g^{0}f^{0}g^{0}f^{0} \simeq g^{0}1_{Y_{0}}f^{0} \simeq 1_{X_{0}} $$ determined by $H^0,K^0$.

I do not know of a counterexample to the idea of avoiding the kind of "conjugation" given above (though it has been given in the dual situation).

Note that this Addendum easily gives a gluing theorem for $n$ subspaces with a common intersection.

I'll add that the idea for this result came from generalising the proof that a homotopy equivalence of spaces (not necessarily base point preserving) induces an isomorphism of homotopy groups.

My memory is that another paper relevant to the question, but to which I do not have easy access, is

Spanier, E. H.; Whitehead, J. H. C. The theory of carriers and S-theory. Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 330–360. Princeton University Press, Princeton, N.J., 1957.

particularly the work on carriers.

Answer by Ronnie Brown for Group extensions with a non-commutative kernel

2013-05-07T13:49:07Z

This should also be linked to Dedecker's notions of cohomology with coefficients in a crossed module, where the usual theory is for the crossed module $K \to Aut(K)$, and also to a paper of Turing linking with identities among relations. See our paper on the Schreier theory, Proceedings Royal Irish Academy 96A (1996) 213-227.. The link to crossed complexes is also stressed in the book Nonabelian Algberaic Topology, Chapter 12; this link allows the use of model category type techniques, such as fibrations of crossed complexes, and tensor products of crossed complexes for calculational purposes.

Answer by Ronnie Brown for Modern Mathematical Achievements Accessible to Undergraduates

2013-05-06T13:49:26Z

There are all sorts of assumptions behind the question, and answers, such as that the solution of famous problems is the test of progress in mathematics. At my first international conference in 1964 I met Stanislaw Ulam, and he mentioned to me: "A young person may think the most ambitious thing to do is to tackle some famous problem or conjecture. But that might distract that person from developing the kind of mathematics most appropriate to them."

In the 1980s I gave a talk to teachers and children on "How mathematics gets into knots" and mentioned prime knots and prime numbers. After the talk, a teacher came up to me and said: "That is the first time anyone in my career has used the word "analogy" in relation to mathematics." I find that tragic! We have to be careful not to fall into: "They ask for bread and we give them stones."

The notion of fractal is well known to the public, but how many mathematics courses give a simple account of the Hausdorff metric, and let students see some of the mathematics behind the fractal notion.

Other scientists would like to know what is new in mathematics, in terms of concepts and ideas. My talk to a Conference on Theoretical Neuroscience in 2003, was well received. it included an email analogy for colimits, as well as ideas on higher dimensional algebra. One participant told me: " That was the first time I had heard a seminar by a mathematician which made any sense!" So this time I managed to get it right!

First year main math courses at University should contain something which excites the imagination. (I am told Physics courses usually have something on current research.) That Euclidean geometry has been out of most syllabi makes this harder, especially to get over the idea of proof. Is it too harsh to say that courses on "Proof" are about how to write clear proofs of boring things? It is good to show proofs of otherwise not so believable things.

In the 20th century, a main contributor to the unity of mathematics has been Category Theory; I feel this is a high order mathematical achievement! See the article Analogy and Comparison for a discussion of analogy in this context.

A simple talk to the first year on cubes of dimension $0$ to $5$, and how to count faces of various dimensions, awakened the interest of a student, who later went on to a PhD. (This was also a talk I have given to 13 year olds: they end up by counting the $2$-dimensional faces of a $5$-dimensional cube.)

There is also a lot to say about the contribution of mathematics over the millennia to science and culture. See an article on Mathematics in Context.

Answer by Ronnie Brown for Generalized Categories for "Higher Homotopy Groupoids"

2013-04-23T14:15:35Z

Here is some background. In 1965 I noticed that the proof of the van Kampen theorem for the fundamental groupoid seemed to generalise to dimension 2, but there was a lack of a suitable gadget, a homotopy double groupoid. I also noticed that a proof due to J.F. Adams that any map $ S^r \to S^n$ for $ r < n $ is inessential (7.6.1 of Topology and Groupoids) should have algebraic consequences, but again there was no appropriate algebraic gadget. Nine years later we had found out a lot about double groupoids and crossed modules, but were still lacking the homotopy double groupoid! Then Philip Higgins and I did a strategic analysis which went as follows:

J.H.C. Whitehead had a subtle theorem on $\pi_2(X \cup_\lambda e^2_\lambda,X,x)$ as a free crossed $\pi_1(X,x)$-module. This was an example, maybe the only then example, of a $2$-dimensional universal property in homotopy theory. (here is a link to an exposition of Whitehead's proof).
If our conjectured $2$-dimensional van Kampen theorem was to be any good it should have Whitehead's theorem as a corollary.
Whitehead's theorem was about relative homotopy groups.
So we should look for homotopy double grouopids in a relative situation, i.e. a space $X$ and a subspace $A$.
The simplest way of doing this we could think of was to consider

as in the above diagram maps of a square into $X$ which takes the edges into $A$ and the vertices to a subset $C$ of $A$ and to take homotopy classes of these rel vertices of the square. The proof that this works and gives a strict double groupoid is not entirely trivial!

To our delight, this went swimmingly, and we were able to prove a $2$-d van Kampen theorem which had Whitehead's theorem as a Corollary. In fact we computed for example $\pi_2(X \cup_f CA,X,x)$ with Whitehead's theorem the case $A$ (not now a subspace) was a wedge of circles.

Another surprise was that the submitted paper was asked to be withdrawn, in order not to embarrass the editor and two international authorities! A request for more information got another referee and a request (order?) to cut the paper by one third. So the final paper had no pictures, and some slicker arguments.

We also managed to work out the results for filtered spaces, and so all dimensions, and these were published in JPAA, 1981. See the book on Nonabelian algebraic topology.

I emphasise that the basic methods were cubical, and we were unlikely to conjecture let alone prove these results using globular or simplicial methods.

I like to think that these methods fulfill the dreams of the topologists of the early 20th century to find higher dimensional versions of the nonabelian fundamental group, since the nonabelian nature of the fundamental group was known to be useful in geometry and analysis.

Over to you, reader, to get such applications!

Added 28 April: The contrast with what are called in the literature "fundamental higher groupoids'' is that:(i) those do not generalise the usual fundamental groupoid since they are not strict, and are just singular complexes; (ii) while they do satisfy some version of what is called the "small simplex theorem" that does not directly imply strict colimit theorems in higher dimensions of a nonabelian type; (iii) the versions of higher groupoids we have worked with are strict structures, are defined non trivially for filtered spaces or $n$-cubes of spaces, and satisfy nonabelian colimit theorems with consequences not so far obtainable by other means. See for example the nonabelian tensor product of groups.

These ideas relate to and are aimed at relative homotopy theory, and $n$-adic homotopy theory, and I hope are seen in low dimensions as relevant to geometric group theory and to geometric topology.

The point is that one needs to evaluate what different approaches do and do not do, to compare and contrast.

See also the question and answer to http://mathoverflow.net/questions/128831/128863#128863 .

Answer by Ronnie Brown for Infinity-categories vs Kan complexes

2013-04-26T21:29:21Z

Recall that Kan's first paper was cubical, but subsequent papers were simplicial as difficulties were found with the standard cubical category, in particular that cubical groups were not Kan, and the realisation of the cartesian product of cubical sets did not have the homotopy type of the product of the realisations. These deficiencies have been overcome by using cubical sets with connections (as developed in work with Higgins) in papers by A. Tonks, (JPAA 81}~(1) (1992) 83--87) and G. Maltsiniotis, (HHA 11~(2) (2009) 309--326.) respectively.

In my own work with Higgins, we have used cubical sets extensively see the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. One major reason for the success of cubical methods is that they can easily express multiple compositions:

Thus in the above diagram, you can ask non mathematicians if they think that there should be some mathematics in which the left hand aquare can be regarded as some kind of composition of the right hand square, and they will happily agree. Fortunately, this language exists, as double categories, and in the case of the cubical singular complex we can also write down such compositions as a composition $[a_{ij}]$ of an array $(a_{ij})$ of cubes. (See Remark 13.1.11 of the above book.) So we have an "algebraic inverse to subdivision" and this can be used to obtain new local-to-global results. The notion of "connections" on cubical sets arose in trying to define "commutative cubes".

The main theorems of the book were envisaged cubically, and eventually realised, but I feel that simplicially or globularly they would not even be thought of. Indeed, this programme was, what should I say, not welcomed by some.

Further, cubes are excellent for homotopies and higher homotopies, because of the rule $I^m \times I^n \cong I^{m+n}$.

So my answer to the question on $\pi_2 X$ and how to see it simplicially is that one should look to the geometry and not try to force an intuition into an uncomfortable mode, especially when a convenient and natural mode is available, or at any rate, the pros and cons should be thought about carefully.

Answer by Ronnie Brown for Are k-spaces named for Kelley?

2013-04-28T10:42:06Z

Just to add a bit to the history, the first time the exponential law was given for Hausdorff k-spaces was I believe in my DPhil thesis, submitted 1961, see PtA available here, which was circulated to the obvious places. In my first paper, (1963), also available here, I wrote:

"It may be that the category of Hausdorff k-spaces is adequate and convenient for all purposes of topology."

In my second paper, (1964), I used the category of Hausdorff spaces and functions continuous on compact subsets, and showed it was what we now called cartesian closed. (My thesis contains an attempt at showing the idea of what we now call monoidal closed, since in my thesis I had lots of internal homs and associated products, usually a tensor.)

I did not understand final topologies at the time and so did not come up with the definition for the non Hausdorff case, but you can find that in my book Topology and Groupoids.

Several people wrote about the non Hausdorff case, but an important application is to fibred exponential laws which were developed by Peter Booth following some ideas sketched by R. Thom. See Booth, Peter I. The section problem and the lifting problem. Math. Z. 121 (1971), 273–287.

I also wonder whether the category defined in

Johnstone, P. T. On a topological topos. Proc. London Math. Soc. (3) 38 (1979), no. 2, 237–271.

is indeed adequate and convenient for all purposes of topology, in particular can cope well with fibred exponential laws, since being a topos is a stronger condition.

There are also the purposes of analysis, for which see

Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997.

So the term "convenient" has had a good run!

Answer by Ronnie Brown for Stratifications and Cohomology Computations

2013-04-24T14:24:47Z

Is the stratification you consider an example of a filtered space, i.e. a space $X$ and a sequence $X_0 \subseteq X_1 \subseteq X_2 \subseteq \cdots $ of subspaces? The algebric topology of these is considered in the book published by the EMS in 2011 as Tract in Mathermstics Vol 15, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. There are also comments about stratifications in Grothendieck's "Esquisse d'un programme" Section 5, which may be relevant, of which a translation is published in a book by Leila Schnepp.

Answer by Ronnie Brown for Giving $Top(X,Y)$ an appropriate topology

2013-04-22T14:55:22Z

@Paul: Paul asks for the motivation: here is my story.

I gave an MSc course on homotopy theory at Liverpool in 1960-61, and was struck then by the nice properties of the category of simplicial sets as against that of topological spaces, thus suggesting the convenience of simplicial sets. My thesis topic then was the algebraic topology of function spaces, and in the process of solving the particular problem I used exponential laws for spaces, simplicial sets, based simplicial stets, chain complexes, simplicial abelian groups, and maybe others. At the end of this work it struck me that the exponential law depended on the product as well as the hom, and I wrote this up as a small introduction.

I also knew that the weak (i.e. k-ified) product has been studied by Whitehead and by Danny Cohen, so it seemed reasonable to try this for the exponewntial law. To my surprise, it all worked well and became the first chapter of my thesis, on the category of Hausdorff k-spaces, which was submitted, and the thesis was reproduced in the old purple Banda and well circulated, e.g. to Princeton.

Writing up this general topology part as papers it all became more expansive and was published as my first two papers, in 1963 and 1964. In writing up the Introduction of the first paper, I speculated: "It may be that the category of Hausdorff k-spaces is adequate and convenient for all purposes of topology." The major properties for conveninece were listed in the second paper, mainly being cartesian closed. I should say that a referee of the initial version had drawn my attention to the important point about cartesian closed, i.e. the usual properties of the product. Also, later workers eliminated the Hausdorff assumption.

For more discussion, see the the ncatlab on convenient categories of topological spaces.

There is also a nice paper of Lawvere discussing the various equivalences between $$(X^Y)^I, (X^I)^Y, X^{Y \times I}$$ in terms of motion and phase spaces, which I will try to find a reference to.

Finally, Spanier's suggestion of quasi-topological spaces is even more convenient, since it is locally cartesian closed, but was rejected mainly because the quasitopologies on the 2-point set formed a class. Maybe Peter Johnstone's "Topological topos" would be adequate and convenient for topology!

Answer by Ronnie Brown for The definition of a CW complex and related notions

2013-04-22T10:19:36Z

Whitehead's definition of CW-complex had a long gestation, and was initially formulated by him as a "membrane complex", see his very original 1941 Annals paper on "On incidence matrices, nuclei and homotopy types", which includes the initial research on what he later called simple homotopy theory. The idea for such complexes was partially to have something coarser than a simplicial complex. Later he developed the notions of adjunction space, and also the general topology; Ioan James said it took him a year to prove his product theorem for CVW-complexes.

Now we can see more clearly that the great advantage of CW-complexes is the inductive definition of the structure, since this allows, as shown by Whitehead, proofs by induction. So the inductive definition is often taken as a starting point.

The skeletal filtration of a CW-complex is also usefully seen as an example of a filtered space, i.e. a space $X$ with an increasing sequence of subspaces $X_0 \subseteq X_1 \subseteq X_2 \subseteq \cdots$. The book advertised here develops algebraic topology at the border between homotopy and homology on the basis of this structure, without singular homology or simplicial approximation, and relating it to Whitehead's key notion of free crossed module (developed 1941-1949), an example of a universal property in $2$-dimensional homotopy theory.

The notion of a space with structure seems to me to be related to Grothendieck's remarks in "Eaquisse d'un programme" Section 5, where he claims the notion of topological space is derived from analysis and is inadequate for geometry. For his purposes, notions of stratifications are crucial. In any case, to specify a topological space one needs some kind of data, and it is not unreasonable to suggest that the invariants we seek should have some structure reflecting that of the data defining the space.

In particular, there are van Kampen type theorems for filtered spaces (as in the above mentioned book) and also for $n$-cubes of spaces, as in work with J.-L. Loday. The latter work is related to classical work on the homotopy theory of $n$-ads, their connectivity theorems, and the determinations of the critical (i.e. first non zero) groups (Blakers-Massey, Barratt-Whitehead). See also the paper by Ellis and Steiner referenced there.

Answer by Ronnie Brown for Compelling evidence that two basepoints are better than one

2010-11-21T10:32:57Z

I came into groupoids by trying to find a new proof of the fundamental group of the circle. It turned out that one could do this using the fundamental groupoid on two base points. Writing the 1968 edition of my book now called `Topology and Groupoids' (T&G) (available on amazon.com and e-version from www.kagi.com) convinced me that all of 1-dimensional homotopy theory was better expressed in terms of groupoids rather than groups, in that one obtained more powerful theorems with simpler proofs. Later results on the fundamental groupoid of orbit spaces (Chapter 11 of T&G) are more awkward to express in terms of groups; this elaborates on the point by Dustin Clausen. See further details below.

Henry Whitehead answered the question of "Why not restrict to CW-complexes with just one vertex?" by considering covering spaces. Philip Higgins gave a considerable generalisation of Grusko's theorem by considering covering morphisms of groupoids, see his 1971 book `Categories and groupoids' available as a TAC Reprint, 2005.

In 1966 I thought about prospective uses of groupoids in higher homotopy theory, and this led over many years to higher dimensional Seifert-van Kampen Theorems, with a range of new nonabelian calculations of second relative homotopy groups and triad homotopy groups (for the latter, see the "nonabelian tensor product of groups"). That sounds relevant to geometric topology!

So one answer to the original question is that the use of groupoids opens new worlds of possibilities.

Actually the idea of `change of base point for the fundamental group' is a bit bizarre: one does not describe a railway timetable in terms of return journeys and change of starting point for these! Why is this still taught to students?

In the end, an aesthetic viewpoint implies more power!

Thanks to those above who give me additional examples.

More information on my page From groups to groupoids.

September 2012: I forgot to add to this answer more information on orbit spaces, with particular reference to "two base points".

Ross Geoghegan in his 1986 review (MR0760769) of two papers by M.A. Armstrong on the fundamental groups of orbit spaces wrote: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." At present, to my knowledge, "Topology and Groupoids" is the only topology text to cover such results.

Consider the action of the cyclic group of order 2, $Z_s$ on the unit circle $S$ by complex conjugation. Take $1$ as base point. The induced action of $Z_2$ on the fundamental group $\pi_1(S,1)$ is $n\mapsto -n$, and the quotient by this action is $Z_2$. But the quotient of $S$ by the action is a semicircle, which is contractible. What has gone wrong?

The problem is there are two fixed points of the action. The quotient of the action of $Z_2$ on the groupoid $\pi_1(S, A)$, where $A$ consists of the points $\pm 1$, is indeed correct.

The point is that a group acting on a space $X$ acts also on the fundamental groupoid $\pi_1 X$. If $X$ is Hausdorff, the action is properly discontinuous, and $X$ has a universal cover, then the fundamental groupoiud of the orbit space $X/G$ is the orbit groupoid of $\pi_1 X$. This is the groupoid expression of Armstrong's results. See Chapter 11 of Topology and Groupoids.

April 21,2013: The book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids gives an account of this new approach to basic algebraic topology at the border between homology and homotopy, without using singular homology theory, or simplicial approximation, but relying on the idea of multiple compositions of cubes. This also allows for results on second relative homotopy groups, results which, being essentially nonabelian, are not obtainable by traditional algebraic topology. It also avoids the "trick" of taking the free abelian group on ordered or oriented simplices in order to define chain groups, and the boundary map.

All this comes from considering the question: if groupoids are useful in $1$-dimensional homotopy theory, how useful can they be in higher homotopy theory? One quickly notices that whereas group objects internal to groups are abelian groups, group objects internal to groupoids are in some sense "more nonabelian" than groups, as are groupoid objects internal to groupoids. So one looks to such objects to model higher homotopy properties: and this has been achieved.

Answer by Ronnie Brown for Is there a "mathematical" definition of "simplify"?

2013-04-05T10:52:57Z

I would like to confuse matters, perhaps, by bringing in a higher dimensional viewpoint. The following type of example was given by John Baez.

The diagram

$$\matrix{||| & |||||\cr ||| & |||||}$$ can be expressed symbolically by $$2\times (3+5)= 2\times 3 + 2 \times 5.$$ But the symbolic "on a line" expression involves many conventions which have to be learned. Which expression is "simpler"? Current computers work serially, on a line, as I understand it, and $2$-dimensional, or higher, rewriting has not been computerised, but is used in knot theory, higher category theory, and higher dimensional group theory.

The term "simplify" depends on the use, as others have observed above. The "simplest" description of a vector space over a given field is its dimension, but that does not mean we want to remove vector spaces from the literature. In my own field, a common statement has been that "groupoids reduce to groups", but the group description in a given case may be more complicated, since it requires choosing base points and trees (Ugh!). See my Grothendieck quote.

This reflects perhaps that mathematical understanding is about understanding structural implications, so "simplicity" depends on the background structure, and the way this is expressed. A "bigger" structure may make things look simpler!

Answer by Ronnie Brown for Anick resolution

2013-04-01T10:06:06Z

In view of Yemon's reference to group cohomology, I would like to mention Graham Ellis' work on "Homological Algebra Programming". The key point is that he constructs free resolutions inductively together with a contracting homotopy: it is the latter that gives the computational aspect.

There is an explanation of some of this in Section 9.3 of the book Nonabelian algebraic topology, in terms of constructing a "home for a contracting homotopy", as against the more traditional "killing kernels", a method which is notably non algorithmic.

The spirit of this derives from Homological Perturbation Theory, in which also the homotopies are crucial.

Answer by Ronnie Brown for Finitely cocomplete categories of compact Hausdorff spaces

2013-03-20T16:14:44Z

The examples given are part of the case for homotopy colimits. For example the pushout of the two maps $$S^1 \leftarrow S^1 \to S^!$$ given by $z\mapsto z^2, z \mapsto z^3$ is not Hausdorff. But the double mapping cylinder $M$ is a nice CW-complex. Amusingly, this is rel;ated to the case for groupoids, where the pushout in groups of the two maps $$ \mathbb Z \leftarrow \mathbb Z \to \mathbb Z$$ is the trefoil group $T$ with generators $x,y$ and relation $x^2=y^3$, but the homotopy pushout in groupoids is the "trefoil groupoid", say $T'$, with two objects $0,1$, generators $x,y$ at $0,1$ respectively and one arrow $\iota :0 \to 1$ conjugating $x^2$ to $y^3$. The advantage of this is that it "separates" the two group generators $x,y$, and of course $T'$ is the fundamental groupoid of $M$ on two base points, one at each end of the cylinder.

Answer by Ronnie Brown for Examples where it's useful to know that a mathematical object belongs to some family of objects

2013-03-04T10:49:04Z

I have to like and suggest ideas connected with the Seifert-van Kampen theorem for the fundamental group and its extensions to groupoids and higher groupoids. An anomaly in traditional approaches, centred on the fundamental group, was that this theorem did not compute the fundamental group of the circle, which is THE basic example in algebraic topology. The reason is the the circle cannot be represented as the union of two path connected sets with path connected intersection. The solution was to use many base points rather than just one, and so to work in the context of groupoids; this dates from 1967 and a fairly recent account is in the book Topology and Groupoids.

This extension led to ideas of using higher groupoids in homotopy theory, and so to define higher homotopy groupoids, with higher order Seifert-van Kampen theorems. By 1984 this led to the idea of a nonabelian tensor product of groups which act on each other, see the bibliography. As an example, if $M,N$ are normal subgroups of the group $P$, then the commutator map $[\;,\;]: M \times N \to P$ is a biderivation and so factors through a universal biderivation, a morphism $\kappa: M \otimes N \to P$. For example if $M=N=P$ then Ker $\kappa$ is isomorphic to $\pi_3SK(P,1)$. Thus taking groupoids seriously in algebraic topology has led to new algebraic ideas.

Answer by Ronnie Brown for Natural transformations as categorical homotopies

2011-09-17T16:17:39Z

Charles Ehresmann had a natty way of developing natural transformations. For a category $C$ let $\square C$ be the double category of commuting squares in $C$. Then for a small category $B$ we can form Cat($B,\square_1 C$), the functors from $B$ to the direction 1 part of $\square C$. This gets a category structure from the category structure in direction 2 of $\square C$. So we get a category CAT($B,C$) of functors and natural transformations. This view makes it easier to verify the law

Cat($ A \times B,C) \cong $Cat($ A, $CAT($B,C$)).

And this method goes over to topological categories as well:

R. Brown and P. Nickolas, ``Exponential laws for topological categories, groupoids and groups and mapping spaces of colimits'', Cah. Top. G\'eom. Diff. 20 (1979) 179-198.

See also Section 6.5 of my book Topology and Groupoids for using the homotopy terminology for natural equivalences, as it was in the first 1968 edition entitled "Elements of Modern Topology" (McGraw Hill).

Answer by Ronnie Brown for First mention of the fundamental bigroupoid of a space?

2013-02-28T23:16:53Z

I feel, pacé Grothendieck, that the use of fundamental groupoid or $n$-groupoid for a strict structure in dimension $1$ and a weak structure in higher dimensions is confusing! But I seem to be out on a limb on this.

In any case, the higher dimensional strict structures have advantages for calculating homotopical invariants, as shown in my papers with Higgins and with Loday. These higher homotopy groupoid structures are known to be well defined for certain structured spaces, in fact for filtered spaces, and for $n$-cubes of spaces. This relates them well to classical homotopical structures, i.e. to relative homotopy groups, and to $n$-ad homotopy groups, and yield new calculations and understanding of of these.

I hope it will be useful to explain the route to this conclusion.

Philip and I defined a strict fundamental double groupoid $\rho(Y,X,x)$ of a pair of pointed spaces in 1974, but Frank Adams' opposition delayed the publication till 1978. I had been looking for a homotopy double groupoid of a space since 1965, but could not do it. In 1974 Philip and I agreed:

Whitehead's subtle theorem (1941-1949), proved using transversality and knot theory, that the crossed module $ \delta: \pi_2(X \cup _\lambda e^2 _\lambda,X,x) \to \pi_1(X,x) $ is free on the $2$-cells was an example of a universal property in $2$-dimensional homotopy theory;
If our conjectured $2$-dimensional van Kampen was true in some form, then it should imply Whitehead's theorem.
Whitehead's theorem involves relative homotopy groups.
We should therefore try to define a homotopy double groupoid in a relative situation.
Given we had little time remaining in Philip's stay at Bangor, we should try the simplest idea, and that seemed to be to consider homotopy classes of maps of a square $I^2$ into $Y$ which mapped the edges into the subspace $X$ and the vertices to the base point. Note that this a symmetric definition, and requires no choice of which edges to be mapped to the base point, as in the usual (and so unaesthetic?) definition of relative homotopy groups.
This was a sensible definition; the work that had already been done by then with Chris Spencer on double groupoids with connections, commutative cubes, and the precise relation of these double groupoids with crossed modules, allowed a proof of the van Kampen theorem and, and given work by Philip on induced crossed modules, a deduction of Whitehead's theorem, and indeed a generalisation to a theorem on $\delta:\pi_2(X\cup CA,X,x)\to \pi_1(X,x)$, with Whitehead's theorem being the case $A$ is a wedge of circles.

This suggests the advantage of a proper strategic analysis, even if delayed!

Note that one of the advantages of strict structures is that the calculation of colimits of them is reasonably clear; in particular, the calculation of colimits of crossed modules is an interesting extension of the calculation of colimits of groups. In algebraic topology, one use of homotopical invariants is to show that objects are not equivalent, and this needs precise answers.

It was then not hard to see the putative extension to filtered spaces; the work was in fact quite hard technically and conceptually, but was completed in 1977, with two CRAS notes published in that year, and the full papers in 1981.

Actually the interest of Whitehead's theorem seems not so much appreciated. It allows one, for the usual representation of the Klein bottle as an identification of a square $\sigma$, to write the more precise

$$\delta \sigma = a+b-a +b $$

instead of the usual $\partial \sigma = 2b$.

In 1982 Loday published a paper giving a definition of a fundamental cat$^n$-group of a pointed $n$-cube of spaces, and showed that such structures modelled all weak, pointed homotopy $n$-types. In a visit of mine to Strasbourg towards the end of 1981, I talked about my work with Philip Higgins, and together Loday and I conjectured a van Kampen type theorem for his cat$^n$-groups. This was proved by 1984 and published in 1987. It allows some quite new calculations of homotopical invariants, e.g. the homotopy $3$-type of $SK(G,1)$, the suspension of an Eileneberg-Mac Lane space.

These theorems have strong limitations, as is only to be expected, but they do allow new determinations and open out new prospects.

So one possible moral is that the concentration on bare topological spaces, without any further structure, is a more restricted endeavour. This fits with Section 5 of Grothendieck's "Esquisse d'un programme", where he argues that the needs of geometry require more than just a topology, and he argues for kinds of stratifications. One way of looking at this is to say that the specification of a space requires some kind of data, and that data will have some kind of structure; so, conjecturally, one should look for invariants of spaces with that kind of structure.

It would be interesting to compare these higher homotopy groupoid structures, and the weak structures in common current parlance, with the vision of the algebraic topologists of the early 20th century for higher dimensional versions of the nonabelian fundamental group. Since the finding in 1932 that Cech's definition of higher homotopy groups, which he submitted to the ICM at Zurich, led to abelian groups, this has seemed to be a mirage.

Answer by Ronnie Brown for Has any attempt been made to classify finite groupoids?

2013-02-27T15:10:32Z

@Fernando: @Todd: I'd just like to add to Todd's remark on the classification of groupoids up to isomorphism. It was early realised that any groupoids is the disjoint union of its connected components; and that given any $cx \in Ob(G)$ for a connected groupoid $G$ ithen $G$ s isomorphic to $G(x) * T$ where $G(x)$ is the vertex, or object group, at $x$ and $T$ is a "tree groupoid", i.e. $T(y,z)$ is a singleton for all $y,z \in Ob(G)$. However this determination depends on first choosing the object $x$ and then for each $ y \ne x$ in $Ob(G)$, choosing an element in $G(x,y)$. So there are lots of choices. As Fernando remarks, a single connected groupoid is up to homotopy "the same as" a group.

However the relation of groupoids to other areas of mathematics is interesting.

Now what the objects of a groupoid add to a group is a kind of "spatial" character. This allows all sorts of new possible interactions beteeen different grouopids, quite unlike those of groups. This is especially relevant to van Kampen type situations. Further, the choices involved in the above determination imply that the classification of diagrams of groupoids does not reduce to the classification of diagrams of groups.

Further, morphisms of groupoids have much more variety than do those for groups: for groupoids we have equivalences, fibrations, covering morphisms (related to actions on sets), quotient morphisms (factor by a normal subgroupoid), universal morphisms (identify objects in some way), orbit morphisms, .... So it is often in the relations between groupoids rather than the classification of single groupoids that we should see the benefit of their use. This reflects the categorical viewpoint.

Answer by Ronnie Brown for Understanding Adjointness of Sheaves in Algebraic Geometry

2013-02-22T10:32:26Z

I used pullbacks and pushouts, or pushforwards, for years (decades!) before realising the nice context in terms of fibrations and cofibrations of categories, and so this context got written up in the paper "Algebraic colimit calculations in homotopy theory using fibred and cofibred categories", with general matter based on notes of Thomas Streicher, and so Benabou, but with a few new simple results.

Thus I find it helpful to see the adjointness result you give as Thm and ask about as a special case of a standard and more general result, when you have, in essence, pullbacks and pushouts, see Proposition 3.6 of our paper. This should illustrate the principle that a more general result may be easier to prove, because one is not confused by the special situation. I have not checked this in your case and would like to hear if it is so!

Answer by Ronnie Brown for What space classifies bundles of K(pi,1)'s?

2013-02-15T11:03:51Z

To put another context to the above comments and answers, recall that a crossed module consists of a morphism $\mu: M \to P$ together with an action of $P$ on the right of the group $M$ satisfying the two axioms

$\mu(m^p)= p^{-1}\mu(m) p$;
$ m^{-1} nm = m^{\mu n}$,

for all $m,n \in M, p \in P$. This definition is due to J.H.C. Whitehead in 1946.

Examples of crossed modules are:

the inner automorphism crossed module $\chi: M \to Aut(M)$ for any group $M$;
the inclusion $M \to P$ of a normal subgroup $M$ of $P$;
the zero map $0: M \to P$ for any right $P$-module $M$;
the induced map $\pi_1(F) \to \pi_1(E)$ for any pointed fibration $F \to E \to B$;

and others!

Any such crossed module has a classifying space $B(M \to P)$ whose homotopy groups are trivial above dimension $2$, and with $\pi_1 \cong Coker \mu$, $\pi_2 \cong Ker \mu$. More details of these facts are in the book EMS Tract 15. Crossed modules are equivalent to group objects in the category of groupoids, and this gives one way of defining the classifying space, using bisimplicial sets, and are conveniently regarded as $2$-dimensional versions of groups, since they model pointed weak homotopy $2$-types. Note that the second homotopy group, even considered as a module over $\pi_1$, is generally but a pale shadow of the homotopy $2$-type.

There is also a Seifert-van Kampen type theorem with values in crossed modules, and this allows some computations of nonabelian second relative homotopy groups. See again the EMS Tract 15.

Answer by Ronnie Brown for A possible generalization of the homotopy groups.

2010-10-04T09:57:53Z

I was told by Brian Griffiths that Fox was hoping to obtain a generalisation of the van Kampen theorem and so continue work of J.H.C Whitehead on adding relations to homotopy groups (see his 1941 paper with that title).

However if one frees oneself from the base point fixation one might be led to consider Loday's cat$^n$-group of a based $(n+1)$-ad, $X_*=(X;X_1, \ldots, X_n)$; let $\Phi X_*$ be the space of maps $I^n \to X$ which take the faces of the $n$-cube $I^n$ in direction $i$ into $X_i$ and the vertices to the base point. Then $\Phi$ has compositions $+_i$ in direction $i$ which form a lax $n$-fold groupoid. However the group $\Pi X_*= \pi_1(\Phi, x)$, where $x$ is the constant map at the base point $x$, inherits these compositions to become a cat$^n$-group, i.e. a strict $n$-fold groupoid internal to the category of groups (the proof is non trivial).

There is a Higher Homotopy van Kampen Theorem for this functor $\Pi$ which enables some new nonabelian calculations in homotopy theory (see our paper in Topology 26 (1987) 311-334).

So a key step is to move from spaces with base point to certain structured spaces.

Comment Feb 16, 2013: The workers in algebraic topology near the beginning of the 20th century were looking for higher dimensional versions of the fundamental group, since they knew that the nonabelian fundamental group was useful in problems of analysis and geometry. In 1932, Cech submitted a paper on Higher Homotopy Groups to the ICM at Zurich, but Alexandroff and Hopf quickly proved the groups were abelian for $n >1$ and on these grounds persuaded Cech to withdraw his paper, so that only a small paragraph appeared in the Proceedings. It is reported that Hurewicz attended that conference. In due course, the idea of higher versions of the fundamental group came to be seen as a mirage.

One explanation of the abelian nature of the higher homotopy groups is that group objects in the category of groups are abelian groups, as a result of the interchange law, also called the Eckmann-Hilton argument. However group objects in the category of groupoids are equivalent to crossed modules, and so are in some sense "more nonabelian" than groups. Crossed modules were first defined by J.H.C. Whitehead, 1946, in relation to second relative homotopy groups. This leads to the possibility, now realised, of "higher homotopy groupoids", Higher Homotopy Seifert-van Kampen Theorems, and the notions of higher dimensional group theory.

Answer by Ronnie Brown for Second homotopy of a torus complement in the 4-sphere

2013-02-14T21:50:35Z

The paper

Martins, João Faria The fundamental crossed module of the complement of a knotted surface. Trans. Amer. Math. Soc. 361 (2009), no. 9, 4593–4630.

looks relevant.

Note that for links in $\mathbb R^3$ it is standard to use the fundamental group and a Seifert-van Kampen Theorem. So for the case in question this paper uses the fundamental crossed module and a $2$-dimensional Seifert-van Kampen Theorem.

Answer by Ronnie Brown for Semidirect product of groupoids

2013-02-15T11:45:22Z

An action of a groupoid on another groupoid was defined in my paper "Groupoids as coefficients" Proc LMS (3) 25 (1972) 413-426, available here, which also uses methods of fibrations of groupooids. The term "coefficients" refers to nonabelian cohomology. This paper uses the term "split extension" instead of semidirect product. The more modern term is used given an action of a group on a groupoid in the book Topology and Groupoids, where it is relevant to the explicit description of orbit groupoids. Note that if a group acts on a space $X$, then it has an induced action on the fundamental groupoid $\pi_1 X$.

Answer by Ronnie Brown for How can I understand the "groupoid" quotient of a group action as some sort of "product"?

2012-02-05T18:45:18Z

The notion of semidirect product $\Gamma \rtimes G$ where $G$ is a group acting on a groupoid $\Gamma$ is set up in Chapter11, Section 11.4, of my book "Topology and Groupoids".

It is used there in connection with studying orbit groupoids, and their relevance to the fundamental groupoid of an orbit space by a group action.

One nice point is that this semidirect product includes the case $\Gamma$ is a discrete groupoid, i.e. essentially a set, when you get what is commonly called the action groupoid. In this case the morphism $p: \Gamma \rtimes G \to G$ is known as a covering morphism of groupoids, and all covering morphisms of $G$ arise in this way.

I feel the use of covering morphisms of groupoids makes for a nice exposition, base point free, of the theory of covering spaces. Such an idea was pointed out for the simplicial case in the 1967 book on simplicial theory by Gabriel and Zisman, was used in the first 1968 edition of my book, and is used in Peter May's 1999 book "A concise course in algebrac topology".

Answer by Ronnie Brown for Free product of categories

2013-02-14T17:49:06Z

As Higgins pointed out in his papers and book, the useful construction for groupoids is what he calls the universal groupoid $U_\sigma(G)$ on a set $Y$ determined by a function $\sigma: Ob(G) \to Y$. Once you have set up a normal form for this, then you have normal forms for free goups and free groupoids and free products of groups and groupoids, where the last are really determined by a pushout on object sets.

These constructions are relevant to topology using the fundamental groupoid on a set of base points and the corrsponding generalisation of the theorem of Seifert-van Kampen.

The inclusion of the category of groups into the category of groupoids has a left adjoint which can be defined using the above universal construction. Also this inclusion of categories preserves colimits of connected diagrams.

Similar considerations presumably apply to (small) categories.

Answer by Ronnie Brown for What are the symmetries of a principal homogeneous bundle?

2013-02-12T15:57:57Z

The OP asks: How can groupoids be used to describe symmetries in this category? Here are some suggestions for starting.

A principal $G$-bundle $E \to B$ can also be desribed as a groupoid $P= EE^{-1}$ over $B$, a construction due to C. Ehresmann. Here $P(b,c)$ is the set of $G$-maps $E_b \to E_{c}$. So one may ask: what for groupoids generalises the well known inner automorphism map $G \to Aut (G)$ for a group $G$?

Now the category $Gpd$ of abstract groupoids is cartesian closed, this is one aspect of the utility of groupoids. We can write the exponential law as a bijection $$Gpd(G \times H,K) \cong Gpd(G, GPD(H,K)).$$ The objects of $GPD(H,K)$ are the morphisms, or functors, $H \to K$ and the arrows of $GPD(H,K)$ are the natural equivalences of functors. In the case of groups, these are just conjugacies of morphisms.

So for any groupoid $G$ there is an endomorphism object $END(G)$ which is a monoid object in groupoids, and this has a maximal subgroup object $AUT(G)$ which is a group object in the category of groupoids. However as shown in the paper available here, group objects in groupoids are equivalent to crossed modules, and the crossed module one obtains by this process is of the form $d: S(G) \to Aut(G)$ where is $S(G)$ is the group of admissible sections $\sigma$ of $G$ as defined by Ehresmann in his paper on topological and differentiable categories. Such a $\sigma $ is a section of say the source map $s$ such that $t\sigma$ is a bijection on $Ob(G)$. These have a multiplication defined by Ehresmann: $\sigma \tau (x)= \sigma (t\tau x) \tau)x)$ (or analogous, depending on conventions conventions). (Such $\sigma$ are called bisections in Mackenzie, K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Volume 213. Cambridge University Press, Cambridge (2005). )

Can one use this framework to study the differentiable, or Lie, case?

I may be able to add more later.

Comment by Ronnie Brown

2013-05-19T14:57:09Z

To add to the last comment: Ehresmann's interest in analysis led him to grouoids which he applied to foliations, and also developed the notion of differential groupoid, now called Lie groupoid, and also topological groupoid, and fibre bundles. He used in essence pseudo groups, and related them to ordered groupoids. See also my article `Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions', [147] on my publication list. By contrast, the notion of groupoid has been notably absent or not much developed in most algebraic topology books.

Comment by Ronnie Brown

2013-05-19T14:21:21Z

By a partial function $f : X \to Y$ I mean a triple consisting of $X,Y, Gr(f)$ where $Gr(f)$ is the graph, a functional subset of $X \times Y$, so that $dom(f), range(f)$ are subsets of $X,Y$ resp. Historically, category arose from algebraic topology, where all functions are total. Ehresmann's approach to category theory arose from analysis, and local-to-global questions, hence his difference in style, and relation to geometry. My definition of "higher dimensional algebra" is the study of algebraic structures with partial operations whose domains are defined by geometric conditions.

Comment by Ronnie Brown

2013-05-17T20:36:23Z

Sorry, there was mistype, and the link is pages.bangor.ac.uk/~mas010/nonabtens.html

Comment by Ronnie Brown

2013-05-17T20:33:52Z

@Arturo: It may be helpful to put the result of Claire Miller in the context of the nonabelian tensor product of groups, see the bibliography at pages.bangor.ac.uk/~masoao/nonabtens.html

Comment by Ronnie Brown

2013-05-16T21:03:11Z

@David: The facts on crossed complexes are in the EMS Tract Vol 15 on "Nonabelian algebraic topology" (2011) advertised on my web site (with pdf) and the EMS web site.

Comment by Ronnie Brown

2013-05-16T20:59:53Z

@David: R.Brown, "Fibrations of groupoids", J. Algebra, 15 (1970) 103-132, gave the first definition, and a paper by Anderson, Bull AMS, 1978 contains the facts on geometric realisations you might need. I use the exact sequence of a fibration of groupoids in my book "Topology and groupoids", which also has a Mayer-Vietoris type sequence in the chapter on covering spaces. Maybe also P.R. Heath, groupoid operations and fibre homotopy equivalences, Math Z. 130 (1973) 207-233, is relevant to your interests.

Comment by Ronnie Brown

2013-05-16T17:19:45Z

I mention that that the case $D,C$ are groupoids is well studied, as special cases of fibrations of crossed complexes, although the groupoid case is simpler. Does that help? References given if needed.

Comment by Ronnie Brown

2013-05-14T06:53:10Z

Model category arguments, and I presume also for triangulated categories, tend to be non explicit on homotopies. So I would expect my answer to mathoverflow.net/questions/130116 could be relevant, since it gives explicit desription of homotopies in an analogous situation, relevant to, indeed used as a basis for a basis for, gluing homotopy equivalances.

Comment by Ronnie Brown

2013-05-11T14:13:42Z

[]:pages.bangor.ac.uk/~mas010/pdffiles/… My paper "Function spaces and product topologies" (1964), available [here][1], showed in the Hausdorff case that the compact-open topology allows for a non symmetric monoidal closed category of topological spaces. The non Hausddorff case was treated later by by Booth and Tillotson(Pacific J Math).

Comment by Ronnie Brown

2013-05-07T14:13:44Z

Personally, I am attracted by those advances which widen perspectives without being too technical. So I am fond of the Lawvere advertisement for the topos of directed graphs, in which the logic is non Boolean, as can be easily explained, to undergraduates and to scientists.

Comment by Ronnie Brown

2013-05-07T13:55:39Z

This is relevant to the answers to mathoverflow.net/questions/127841/128362#128362 I mentioned there Peter Johnson'e paper on a "Topologiacal Topos", and I notice that this idea is endorsed by Gaunce in his final section of the Appendix.

Comment by Ronnie Brown

2013-05-06T21:57:49Z

@darij: I agree with the tenor of this answer. I have read a book on Quantum Physics with the disclaimer that it cannot mention the thousands of physicists who had contributed to the advancement of the area under discussion. The advancement of mathematics does also require a broad front, and this general advance needs to be brought to the attention of students. How to do this successfully, in the context of an examination system, is not so clear. A feature of our Maths in Context course was the wide variety of project topics chosen by the students, many not on our lists.

Comment by Ronnie Brown

2013-05-06T19:58:55Z

@John: thanks John. Corrected. The conference was in Syracuse, Sicily, in honour of Archimedes, and Sierpinski and Ulam stayed at the posh hotel; but Stan came down regularly to chat with the rest!

Comment by Ronnie Brown

2013-05-02T10:20:25Z

@Bill: Great to have your response! I wonder about the Peano curve as a pathology. It is not smooth. But then neither is Brownian motion. Should we regard Brownian motion as continuous? I recall Jim EElls (I think it was he) telling of discussing in a bar with a colleague about wild orbits, and the barman interrupted to say he knew a lot about those, as he had been a transformer engineer! Again, a topological topos allows for continuity (smoothness?) of functions with variable domain, such as the solutions of differential equations with parameters, e.g. $x \mapsto \log(x + t)$.

Comment by Ronnie Brown

2013-04-29T09:58:43Z

@Xander: Great: continue to follow up your ideas, who knows where they may lead. Chris Spencer once remarked that it was a peculiar subject, since once one had got rid of the gangue, it all worked out better than one could expect!