Here I list some ideas I am slowly pursuing, sometimes in collaboration with others. new link
The Lie fundamental bigroupoid of a finite dimensional manifold
Joint work with Andrew Stacey.
Outline: In my thesis I define a topological fundamental bigroupoid of a (locally well-behaved) topological space. Since the fundamental groupoid of a fin. dim. manifold can be made into a Lie groupoid, a natural idea is to form the joint generalisation of these ideas. For a description of the fundamental bigroupoid see topological fundamental bigroupoid.
One issue is this: which smooth paths do I use? One obvious solution is piecewise smooth, but Andrew tells me they do not form a nice space. The other obvious solution is paths with sitting instants, but then some of the approaches in the topological case will need some new tricks to make them work. Above all, the ‘obvious’ operations on paths (e.g. concatenation) should all be smooth.
The other things to do are: the manifold of 2-arrows and smoothness of structure maps - source-target-compositions? and smoothness of structure maps - associator-etc?.
Localizing a class of arrows to equivalences
Given a small category and a class of arrows , there is a a small category and a universal functor such that sends arrows in to isomorphisms. This is just from Gabriel-Zisman’s Calculus of fractions and homotopy theory.
Given a bicategory and a class of 1-arrows admitting a bicategory of fractions, we have the analogous situation for bicategories, where elements of are sent to equivalences.
But what about taking a small category and localizing it at a class of arrows to a bicategory, such that those arrows in the class are sent to equivalences in the bicategory, and this construction is universal in the appropriate way?
Well, every small category is a bicategory in a trivial way, so the previous construction should work, and it should result in a -category (a locally groupoidal bicategory). Did you want something more? —Toby
Ah, but the localisation of a general bicategory at a general class of 1-arrows has never been constructed. Pronk’s axioms for fractions hold for a 1-category precisely if Gabriel-Zisman’s do, and that is too strong. I was thinking more along the lines of: the localisation has as 1-arrows the paths in such-and-such a graph, and 2-arrows just enough to enforce the axioms of a bicat, weakly invertible arrows that should be, and functoriality of the inclusion of the original category. -David
You mean that Pronk's requirements for to admit a bicategory of fractions are only satisfied (when is a -category) when the resulting is again a -category (at least up to equivalence)?
On a related note, is Pronk's paper (or at least a precise statement of the theorem, including the requirements on ) available online anywhere? I can't find it, and getting to a library is inconvenient for me, so I haven't been able to check them for myself. —Toby
Here is Pronk’s paper. -Mathieu
A 1-category satisfies the axioms for a bicategory of fractions if and only if it satisfies the axioms for a category of fractions. (This is Remark 17 in Pronk’s paper, which you will have probably already seen) I haven’t run through section 2.3 where she constructs the 2-cells and applied it to the case when we start with a 1-category. One reason why I like anafunctors now is that they do away with the messy ‘2-cells as equivalence classes of pasting diagrams’ treatment. —David
Thanks, Mathieu. I've looked at it before, but now I'll look at it again and then say something intelligent. —Toby
Given a 1-category , the functor (Gabriel-Zisman localization) satisfies EF1 and EF2 of proposition 24 in Pronk’s article, but not necessarily EF3. But, as Matthieu pointed out in a comment here, this doesn’t mean that is not equivalent to (Pronk localization) in general.
(Deleted some of my own stupid comments) —David
Since is in this case a (2,1)-category, to show it is equivalent to a 1-category all we need to do is show that the automorphism group of a given 1-cell is trivial. Using section 2.3 of Pronk’s article, if
(w,f):A \leftarrow M \to B
is a 1-cell is , and is a representative for a 2-cell from to itself, it is equivalent to the identity 2-cell, represented by . Hence is (equivalent to) a 1-category.
OK, I understand now, having reread Pronk and also seen the discussion at category of fractions. You can probably remove my remarks here. —Toby
Clearly the 1-arrows of the 2-localization will be finite length zig-zags where the backward pointing arrows are in . The trick is to introduce the minimal number of 2-arrows so that we get
- a pseudofunctor with its coherence data
- invertible 2-arrows expressing the fact is an equivalence
- universal properties
One idea is to try the 2-localization so that the result is a (2,1)-category, i.e. all 2-arrows are invertible. The associated 1-category given by sending the hom-groupoids to their connected components will then be the ordinary localization of .