Dear Peter, You wrote: -------------------------------- I reworked that theory from scratch when writing ``A concise course in algebraic topology''. Chapter 3 (pp21-32) does covering spaces, covering groupoids, the orbit category and the various equivalences of categories among them. I like it, but that chapter is maybe the main reason that my book is less popular than others: non-categorical types find it too difficult for young minds to absorb the first time around. -------------------------------- It seems to me that you give a complicated route via the universal cover to the inverse equivalence from GpdCov(\pi X) to TopCov(X), which assumes connectivity and so requires a choice of base points. My account starts with any covering morphism q: G \to \pi_1 X of groupoids and gives precise local conditions on X for there to be a `lifted topology' on Ob(G) which makes it a covering space of X with fundamental groupoid canonically isomorphic to G. No connectivity is assumed, which makes it useful for discussing coverings of fundamental groupoids of non connected topological groups. It has other uses, such as topologising \pi_1 X. I now find something quite unintuitive, even bizarre, in any emphasis on `fundamental groups and change of base point': it is like giving railway schedules in terms of return journeys and change of start points. The later editions of my book also give a full account of orbit spaces and orbit groupoids under the action of a group, giving conditions for the fundamental groupoid of the orbit space to be naturally isomorphic to the orbit groupoid of the fundamental groupoid. Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I'm eclectic, and prefer closer contact with the real world of existing applications. There the overwhelming majority of the literature uses universal covers as usually constructed. I didn't go into it, but the dependence of that on the basepoint is also ephemeral: you get a universal covering space functor from the fundamental groupoid to such coverings easily enough. That is also used in applications (quite recently by Kate Ponto in work on fixed point theory). In any case, I don't place the emphasis you do on this matter, which I regard as minor from the point of view of algebraic topology. Peter On 6/2/10 2:03 AM, Ronnie Brown wrote:
Dear Peter,
You wrote: -------------------------------- I reworked that theory from scratch when writing ``A concise course in algebraic topology''. Chapter 3 (pp21-32) does covering spaces, covering groupoids, the orbit category and the various equivalences of categories among them. I like it, but that chapter is maybe the main reason that my book is less popular than others: non-categorical types find it too difficult for young minds to absorb the first time around. --------------------------------
It seems to me that you give a complicated route via the universal cover to the inverse equivalence from GpdCov(\pi X) to TopCov(X), which assumes connectivity and so requires a choice of base points.
My account starts with any covering morphism q: G \to \pi_1 X of groupoids and gives precise local conditions on X for there to be a `lifted topology' on Ob(G) which makes it a covering space of X with fundamental groupoid canonically isomorphic to G. No connectivity is assumed, which makes it useful for discussing coverings of fundamental groupoids of non connected topological groups. It has other uses, such as topologising \pi_1 X.
I now find something quite unintuitive, even bizarre, in any emphasis on `fundamental groups and change of base point': it is like giving railway schedules in terms of return journeys and change of start points.
The later editions of my book also give a full account of orbit spaces and orbit groupoids under the action of a group, giving conditions for the fundamental groupoid of the orbit space to be naturally isomorphic to the orbit groupoid of the fundamental groupoid.
Ronnie
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Ronnie and Peter, In the applications of algebraic topology to topology, where do the ‘basepoints’ originate? From my (regrettably too few) contacts with algebraic topologists I gleaned the following: 1. (S. Eilenberg) The base point is the residue of a collapsed subspace, which results, for example, in constructing a model of the 2-sphere by collapsing the boundary of a 2-ball. 2. (B. Eckmann) The pairs, space/subspace (whose homology is often studied) can be usefully generalized to arbitrary maps as objects, not just inclusion maps. 3. (R. Swan) A construction is usually not functorial if one of its steps involves complementation of subobjects; but collapsing subobjects retains nearly the same information, yet is functorial. 4. (M. Artin and G. Wraith) An important refinement of the morphism category of 2. above involves ‘gluing’ along a left-exact functor between two categories, a special ‘comma’ category construction that in fact always yields a topos if the original categories are toposes. For example, the inverse image functor i of a grounding of one topos over another yields in this way a topos whose objects are maps i(S) à E. 5. (P. Freyd) Under the name of sconing the geometrical construction of 4. is very useful in case the objects S of the base topos deserve to be called ‘discrete’. (Ronnie B. points out that this sort of category is the natural domain of the fundamental groupoid.) 6. Suppose a topos (of spaces) is locally connected over another one (of discrete spaces). That means that the inverse image functor i (itself the left adjoint of a points functor) has its own further left adjoint p counting connected components. Then the constructions of 4., 5., yield a result which is again locally connected; the extended p assigns to any Aà E the pushout E/A with Aà i p A. In the spirit of 3. I think of E/A as the exterior of A. This construction is clearly a left adjoint and hence co-continuous in contrast to the construction which merely collapses any A to a point (with which it agrees in case A has exactly one component). Here is a proposed application of the construction of 4., 5, 6., to geometric analysis, serving e.g. as a refutation to the supposed ubiquity of rings without unit: The easy notion of support for covariant quantities like measures is concerned with domain of dependence: An element of M(E) might come from an element of M(A) via A à E and hence be supported on A. Also for contravariant quantities we need not make an abusive use of the properties of minus and zero. A function on X ‘of compact support’ may be interpreted as one that does not depend on the large part L which is remote from some small part K of interest; here K union L = X. The complements of such K are to be inoperative in the variation of such particular functions. But even the line has two ends so that constancy on the components of L is a more functorial condition on functions. If the codomain space R has certain algebraic structure, then R(X,L) = R^(X/L), the exponential space of functions on the indicated pushout enjoys all the same algebraic structure, as does the colimit over all large remote L in X (these being filtered). Of course, this construction R(X/infinity) is functorial only for proper maps X à Y, i.e. those whose inverse image preserves the large remoteness. The covariant dependency of the dual space Hom (R (X/infinity), R) of functionals is likewise only along proper maps, in contrast to that of the smaller space M(X)=Hom(R^X,R) of functionals that have to integrate all functions of the category. Best wishes,Bill
Date: Wed, 2 Jun 2010 08:41:58 -0500 From: may@math.uchicago.edu To: ronnie.profbrown@btinternet.com CC: jds@math.upenn.edu; categories@mta.ca Subject: categories: Re: covering spaces and groupoids
I'm eclectic, and prefer closer contact with the real world of existing applications. There the overwhelming majority of the literature uses universal covers as usually constructed. I didn't go into it, but the dependence of that on the basepoint is also ephemeral: you get a universal covering space functor from the fundamental groupoid to such coverings easily enough. That is also used in applications (quite recently by Kate Ponto in work on fixed point theory). In any case, I don't place the emphasis you do on this matter, which I regard as minor from the point of view of algebraic topology.
Peter On 6/2/10 2:03 AM, Ronnie Brown wrote:
Dear Peter,
You wrote: -------------------------------- I reworked that theory from scratch when writing ``A concise course in algebraic topology''. Chapter 3 (pp21-32) does covering spaces, covering groupoids, the orbit category and the various equivalences of categories among them. I like it, but that chapter is maybe the main reason that my book is less popular than others: non-categorical types find it too difficult for young minds to absorb the first time around. --------------------------------
It seems to me that you give a complicated route via the universal cover to the inverse equivalence from GpdCov(\pi X) to TopCov(X), which assumes connectivity and so requires a choice of base points.
My account starts with any covering morphism q: G \to \pi_1 X of groupoids and gives precise local conditions on X for there to be a `lifted topology' on Ob(G) which makes it a covering space of X with fundamental groupoid canonically isomorphic to G. No connectivity is assumed, which makes it useful for discussing coverings of fundamental groupoids of non connected topological groups. It has other uses, such as topologising \pi_1 X.
I now find something quite unintuitive, even bizarre, in any emphasis on `fundamental groups and change of base point': it is like giving railway schedules in terms of return journeys and change of start points.
The later editions of my book also give a full account of orbit spaces and orbit groupoids under the action of a group, giving conditions for the fundamental groupoid of the orbit space to be naturally isomorphic to the orbit groupoid of the fundamental groupoid.
Ronnie
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Bill, The base point is important in algebraic topology for many reasons. The category of pointed spaces can be understood as the first step toward "embedding" the category of spaces into something like an additive category (for example, into the category of spectra). An additive category is pointed. The base point of a space is playing the role of the null element. A wedge of pointed spaces is their "sum", and a smash product of two pointed spaces is their tensor product. By a theorem of Kan, the model category of pointed connected spaces is Quillen equivalent to the model category of (simplicial) groups. The category of groups is not additive, since a group is not abelian, but it is nearly so. The category of (simplicial) groups is a homotopy variety of algebras, hence also the category of pointed connected spaces. The pointed circle is the natural generator of the category of pointed connected spaces. Every pointed connected space is a homotopy sifted colimit of bouquets of circles. The category of pointed n-connected spaces is a homotopy variety of algebras for every n geq 0. The pointed (n+1)-sphere is the natural generator. Best wishes, André -------- Message d'origine-------- De: categories@mta.ca de la part de F. William Lawvere Date: jeu. 03/06/2010 14:14 À: may@math.uchicago.edu; ronnie.profbrown@btinternet.com Cc: jds@math.upenn.edu; categories Objet : categories: Re: covering spaces and groupoids Dear Ronnie and Peter, In the applications of algebraic topology to topology, where do the 'basepoints' originate? From my (regrettably too few) contacts with algebraic topologists I gleaned the following: 1. (S. Eilenberg) The base point is the residue of a collapsed subspace, which results, for example, in constructing a model of the 2-sphere by collapsing the boundary of a 2-ball. 2. (B. Eckmann) The pairs, space/subspace (whose homology is often studied) can be usefully generalized to arbitrary maps as objects, not just inclusion maps. 3. (R. Swan) A construction is usually not functorial if one of its steps involves complementation of subobjects; but collapsing subobjects retains nearly the same information, yet is functorial. 4. (M. Artin and G. Wraith) An important refinement of the morphism category of 2. above involves 'gluing' along a left-exact functor between two categories, a special 'comma' category construction that in fact always yields a topos if the original categories are toposes. For example, the inverse image functor i of a grounding of one topos over another yields in this way a topos whose objects are maps i(S) à E. 5. (P. Freyd) Under the name of sconing the geometrical construction of 4. is very useful in case the objects S of the base topos deserve to be called 'discrete'. (Ronnie B. points out that this sort of category is the natural domain of the fundamental groupoid.) 6. Suppose a topos (of spaces) is locally connected over another one (of discrete spaces). That means that the inverse image functor i (itself the left adjoint of a points functor) has its own further left adjoint p counting connected components. Then the constructions of 4., 5., yield a result which is again locally connected; the extended p assigns to any Aà E the pushout E/A with Aà i p A. In the spirit of 3. I think of E/A as the exterior of A. This construction is clearly a left adjoint and hence co-continuous in contrast to the construction which merely collapses any A to a point (with which it agrees in case A has exactly one component). Here is a proposed application of the construction of 4., 5, 6., to geometric analysis, serving e.g. as a refutation to the supposed ubiquity of rings without unit: The easy notion of support for covariant quantities like measures is concerned with domain of dependence: An element of M(E) might come from an element of M(A) via A à E and hence be supported on A. Also for contravariant quantities we need not make an abusive use of the properties of minus and zero. A function on X 'of compact support' may be interpreted as one that does not depend on the large part L which is remote from some small part K of interest; here K union L = X. The complements of such K are to be inoperative in the variation of such particular functions. But even the line has two ends so that constancy on the components of L is a more functorial condition on functions. If the codomain space R has certain algebraic structure, then R(X,L) = R^(X/L), the exponential space of functions on the indicated pushout enjoys all the same algebraic structure, as does the colimit over all large remote L in X (these being filtered). Of course, this construction R(X/infinity) is functorial only for proper maps X à Y, i.e. those whose inverse image preserves the large remoteness. The covariant dependency of the dual space Hom (R (X/infinity), R) of functionals is likewise only along proper maps, in contrast to that of the smaller space M(X)=Hom(R^X,R) of functionals that have to integrate all functions of the category. Best wishes,Bill [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
F. William Lawvere -
Joyal, André -
Peter May -
Ronnie Brown