On 5/30/10 12:50 PM, jim stasheff wrote:
Peter May wrote:
DeTeXing an exercise I routinely assign, here is an example of an isomorphism of categories that is not `accidental' in Peter Johnstone's sense and is always used in practice as an isomorphism and not merely an equivalence.
The fundamental theorem of Galois theory:
Let G = Gal(E/F) be the Galois group of a finite Galois extension E/F. Define an isomorphism of categories between the category of intermediate fields F\subset K\subset E and field maps K >--> L that fix F pointwise and the category of orbits G/H and G-maps between them.
and an isomprhic category of coverings spaces such that...
jim No-no, that is in fact the very next exercise:
Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
If you take a category connCov_*(X) of _pointed_ connected covering spaces of a pointed well-behaved space X, then this is a preorder (at most one arrow between any two objects), and this is equivalent to the poset Sub(G) of subgroups of G = pi_1(X). However, Sub(G) is isomorphic to the reflection of connCov_*(X) into the category of posets. Without the presence of basepoints, it is, as Ronnie Brown would say, more natural to use groupoids. David
No-no, that is in fact the very next exercise:
Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).)
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Peter May wrote: --------------------------- Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).) -------------------------- I would like to put in a case here for the groupoid approach ( see `Elements of modern topology' (1968), and subsequent editions; got the idea from Gabriel/Zisman, so not entirely idiosyncratic). If TopCov(X) is the category of covering spaces of X, and X admits a universal cover, then the fundamental groupoid functor \pi induces an equivalence of categories \pi: TopCov(X) \to GpdCov(\pi X) to the category of groupoid covering morphisms of \pi X. This seems to me to be the most intuitive version - a covering map is modelled by a covering morphism. I prefer the proof in this version, since it does not involve choices of base point, and allows the non connected case. It also allows one to discuss the case X is a topological group and to look at topological group covering maps. (Brown/Mucuk, Math ProcCamb Phil Soc 1994, following up ideas of R.L. Taylor). The notion of covering morphism of groupoids goes back to P.A. Smith (Annals, 1951), called a regular morphism, and nowadays a discrete fibration, I think. Is there an analogous version for Galois theory? Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Is there an analogous version for Galois theory?
In SGA 1 A.G. shows an equivalence C ~= GpoidActions(\pi(C)) where C is a category with certain axioms (that he callls Galoisiene) and \pi(C) is the fundamental groupoid of C (its objects are the set I of fiber functors of C, and its vertice groups are profinite, discrete only in case of existence of universal covering), and GpoidActions(\pi(C)) is the category of families indexed by I with an action of \pi(C). This includes as examples both the case of covering spaces and classical Galois Theory (Artin theory with the algebraic closure) I imagine that the category GpoidActions(\pi(C)) should be equivalent to GpdCov(\pi X) in a general abstract setting. Subsequently, this theory was extended and generalized in a well determined direction (progroupoids, localic groupoids, localic progroupoids) in several steps by A.G. himself, Moerdiejk, Bunge, Dubuc and Joyal-Tierney. Other authors extended the basic theory (presence of universal covering and discrete groups) in different directions. e.d. Ronnie Brown wrote:
Peter May wrote: --------------------------- Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).) --------------------------
I would like to put in a case here for the groupoid approach ( see `Elements of modern topology' (1968), and subsequent editions; got the idea from Gabriel/Zisman, so not entirely idiosyncratic). If TopCov(X) is the category of covering spaces of X, and X admits a universal cover, then the fundamental groupoid functor \pi induces an equivalence of categories
\pi: TopCov(X) \to GpdCov(\pi X)
to the category of groupoid covering morphisms of \pi X. This seems to me to be the most intuitive version - a covering map is modelled by a covering morphism. I prefer the proof in this version, since it does not involve choices of base point, and allows the non connected case. It also allows one to discuss the case X is a topological group and to look at topological group covering maps. (Brown/Mucuk, Math ProcCamb Phil Soc 1994, following up ideas of R.L. Taylor).
The notion of covering morphism of groupoids goes back to P.A. Smith (Annals, 1951), called a regular morphism, and nowadays a discrete fibration, I think.
Is there an analogous version for Galois theory?
Ronnie Brown
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
David Roberts -
Eduardo J. Dubuc -
Peter May -
Ronnie Brown