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1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeMay 28th 2010
   • (edited May 28th 2010)
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexLittle bit of housekeeping at [[linear mapping spaces]] and [[manifold structure of mapping spaces]]. Changed the way that the chart map is defined at [[davidroberts:smooth fundamental bigroupoid]]. I think that this method would work for a more general source space. What one would need is a smooth space $N$ with a boundary $\partial N$ so that $\partial N$ split into "source" and "target" and so that the inclusion $\partial N \to N$ admitted a smooth retraction in a neighbourhood of $\partial N$. The part of the construction under question is the following. One has a smooth map $\sigma \colon N \to M$ and so, by restriction, a smooth map $\partial \sigma \colon \partial N \to M$. Then one has a deformation of $\partial \sigma$, say $\alpha$, and one would like deform $\sigma$ to a map $\tau \colon N \to M$ so that $\partial \tau = \alpha$. The way this is done is by using the fact that $M$ is a smooth manifold and so one can start "dragging" points around on it. Basically, for pairs of "close" points, one can define a diffeomorphism taking the first to the second, and do this smoothly in all pairs of points. So applying these diffeomorphisms we drag $\partial \sigma$ to $\alpha$, and since they are diffeomorphisms of $M$ then as we drag $\partial \sigma$ then the rest of $\sigma$ follows along behind. At the moment, though, this is only spelled out for $N = I$.

   Little bit of housekeeping at linear mapping spaces and manifold structure of mapping spaces.

   Changed the way that the chart map is defined at smooth fundamental bigroupoid (davidroberts). I think that this method would work for a more general source space. What one would need is a smooth space $N$ with a boundary $\partial N$ so that $\partial N$ split into “source” and “target” and so that the inclusion $\partial N \to N$ admitted a smooth retraction in a neighbourhood of $\partial N$.

   The part of the construction under question is the following. One has a smooth map $\sigma \colon N \to M$ and so, by restriction, a smooth map $\partial \sigma \colon \partial N \to M$. Then one has a deformation of $\partial \sigma$, say $\alpha$, and one would like deform $\sigma$ to a map $\tau \colon N \to M$ so that $\partial \tau = \alpha$. The way this is done is by using the fact that $M$ is a smooth manifold and so one can start “dragging” points around on it. Basically, for pairs of “close” points, one can define a diffeomorphism taking the first to the second, and do this smoothly in all pairs of points. So applying these diffeomorphisms we drag $\partial \sigma$ to $\alpha$, and since they are diffeomorphisms of $M$ then as we drag $\partial \sigma$ then the rest of $\sigma$ follows along behind.

   At the moment, though, this is only spelled out for $N = I$.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 28th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexThanks, Andrew. I was thinking recently (today, actually) about how this would end up as a paper. Assume first we actually write some nonzero number of papers. Is it one on the topological bigroupoid, and one on the smooth one? Or one big paper on both? That would make it a little bit easier to put in the result that the canonical weak 2-functor $\Pi_2^{Lie}(M) \hookrightarrow \Pi_2^{top}(M)$ is a weak equivalence of topological bigroupoids, rather than making the poor reader try to work from two papers at once. Note that this result shows that the source fibre, which is a Lie groupoid over $M$, is 2-connected as a topological groupoid, something I'm very interested in, and was one of the main reasons for my thesis. Or do I play the game that the funding bodies in Australia like and maximise my number of papers (say by doing three - proving the equivalence in the third!) I'm being a bit silly, owing to the lateness of the hour, but more seriously, how do you see this project ending up? Should we discuss this via email? Ideally I'd like to prove a result similar to one in my thesis: the source fibre of a 'submersive locally weakly discrete' Lie bigroupoid is a smooth 2-covering space of the object manifold. The jargon in quotes is something that holds for the topological fundamental bigroupoid of a semi-locally contractible space (I prefer $\infty$-well-connected, but I will argue that point elsewhere) giving the 2-connected 2-covering space from my thesis.

   Thanks, Andrew. I was thinking recently (today, actually) about how this would end up as a paper. Assume first we actually write some nonzero number of papers. Is it one on the topological bigroupoid, and one on the smooth one? Or one big paper on both? That would make it a little bit easier to put in the result that the canonical weak 2-functor $\Pi_2^{Lie}(M) \hookrightarrow \Pi_2^{top}(M)$ is a weak equivalence of topological bigroupoids, rather than making the poor reader try to work from two papers at once. Note that this result shows that the source fibre, which is a Lie groupoid over $M$, is 2-connected as a topological groupoid, something I’m very interested in, and was one of the main reasons for my thesis. Or do I play the game that the funding bodies in Australia like and maximise my number of papers (say by doing three - proving the equivalence in the third!) I’m being a bit silly, owing to the lateness of the hour, but more seriously, how do you see this project ending up? Should we discuss this via email?

   Ideally I’d like to prove a result similar to one in my thesis: the source fibre of a ’submersive locally weakly discrete’ Lie bigroupoid is a smooth 2-covering space of the object manifold. The jargon in quotes is something that holds for the topological fundamental bigroupoid of a semi-locally contractible space (I prefer $\infty$-well-connected, but I will argue that point elsewhere) giving the 2-connected 2-covering space from my thesis.

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeMay 28th 2010
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItex> Should we discuss this via email? I'm increasingly finding that if someone sends me an email that I don't have to respond to right there and then, then I frequently don't get round to responding at all! We can shift to email later when we get to the point where we do want to consider our strategy in the Game (since it doesn't do to reveal ones strategy publicly!), but for now there may be benefit from keeping it in the open. Number of papers can be quite fluid, even when they're being written! I've had one paper effectively cannibalised by another, whilst a third paper underwent fission. So let's start with actual content. Also, it may become clear that there's certain parts of the project where I've not had any input whatsoever, in which case it would make sense to put them in one paper, and the bits where I've helped in another. So let's get to content. At the moment, I'm a bit frog-like in this project: I don't have the bird's eye view of the landscape. (I'm a bit of a frog by inclination.) What I'm thinking about is how to construct the darn thing, at least in the smooth case, and have not so clear a picture as to why one might want to! This is presumably all covered in your thesis, which I should now go and read, I guess, but if there's a quick explanation that you could easily give, then that would help too (think of it as a first draft of the introduction!). In particular, in your scheme for maximising the number of papers, you suggest one on the topological, one on the smooth, and one on the equivalence. Ignoring the paper question, isn't the topological stuff already in your thesis? So wouldn't that paper be the published version of (part of) your thesis? In which case, it almost certainly shouldn't be joint and should be separate from the other(s). The smooth paper, on the other hand, could start with the more general result on mapping spaces that I outlined above, and then specify to the paths for the groupoid. The equivalence between the two fits better in the smooth than the topological, since people are used to specialising from continuous to smooth, but less so with expanding from smooth to continuous. (I'm going to move this into the atrium as this isn't about nlab pages anymore)

   Should we discuss this via email?

   I’m increasingly finding that if someone sends me an email that I don’t have to respond to right there and then, then I frequently don’t get round to responding at all! We can shift to email later when we get to the point where we do want to consider our strategy in the Game (since it doesn’t do to reveal ones strategy publicly!), but for now there may be benefit from keeping it in the open.

   Number of papers can be quite fluid, even when they’re being written! I’ve had one paper effectively cannibalised by another, whilst a third paper underwent fission. So let’s start with actual content. Also, it may become clear that there’s certain parts of the project where I’ve not had any input whatsoever, in which case it would make sense to put them in one paper, and the bits where I’ve helped in another.

   So let’s get to content. At the moment, I’m a bit frog-like in this project: I don’t have the bird’s eye view of the landscape. (I’m a bit of a frog by inclination.) What I’m thinking about is how to construct the darn thing, at least in the smooth case, and have not so clear a picture as to why one might want to! This is presumably all covered in your thesis, which I should now go and read, I guess, but if there’s a quick explanation that you could easily give, then that would help too (think of it as a first draft of the introduction!).

   In particular, in your scheme for maximising the number of papers, you suggest one on the topological, one on the smooth, and one on the equivalence. Ignoring the paper question, isn’t the topological stuff already in your thesis? So wouldn’t that paper be the published version of (part of) your thesis? In which case, it almost certainly shouldn’t be joint and should be separate from the other(s). The smooth paper, on the other hand, could start with the more general result on mapping spaces that I outlined above, and then specify to the paths for the groupoid. The equivalence between the two fits better in the smooth than the topological, since people are used to specialising from continuous to smooth, but less so with expanding from smooth to continuous.

   (I’m going to move this into the atrium as this isn’t about nlab pages anymore)

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 29th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>(I'm going to move this into the atrium as this isn't about nlab pages anymore) thanks, I probably needed some air anyway :) >What I'm thinking about is how to construct the darn thing, at least in the smooth case, and have not so clear a picture as to why one might want to! This is presumably all covered in your thesis, which I should now go and read, I guess, but if there's a quick explanation that you could easily give, then that would help too (think of it as a first draft of the introduction!). well, here's a reeal quick attempt. I should probably expand this into something worthy of a blog post, if the cafe is interested. _______ The universal covering space (of a pointed space/manifold) has as its underlying set the source fibre of the fundamental groupoid. The trick is just to turn it into a space or manifold, and this means you need to make $\Pi_1(X)$ into a topological or Lie groupoid. What's particularly nice is that the functor $Diff \to Top_{wc} \stackrel{\Pi_1}{\to} Gpd$, for the sort of spaces we are interested in ([[well-connected space]]s), lifts first to $TopGpd$ and then to $LieGpd$ through the obvious forgetful functors. Taking the source fibre of a pointed groupoid is then functorial wrt strictly pointed functors, so the whole construction is functorial. It then obvious* that the 2-connected cover of a space/manifold should be the source fibre of a suitably topologised/smoothified fundamental bigroupoid. I did the topological case in my thesis, but there is at least one other functorial construction of a 2-connected cover of a topological space, namely via simplicial sets and the functorial n-truncations available there. Aside from providing concrete examples of non-bundle-gerbe 2-bundles, of which there was a paucity in the literature, the whole construction was meant to be a signpost pointing to how to do the smooth case. Now of course the functor $Diff \stackrel{\Pi_2^{Lie}}{\to} Bigpd$ (the 1-category of bigroupoids) doesn't factor through $Top_{wc} \stackrel{\Pi_2^{top}}{\to} Bigpd$ strictly, but it does up to equivalence. My thesis constructed the lift of the latter functor to $TopBigpd$, but really I want to lift the former to $LieBigpd$. Then for pointed spaces we get strictly pointed 2-functors and so we can describe a functor $X \mapsto X^{(2)}$ associating to a space the 2-connected cover, and using your work, the same for pointed manifolds. One potential application is to, as I did in my thesis for the topological case, reverse engineer the definition of smooth 2-covering space (actually this follows almost formally, but it's a good idea to have an example around to help). Then I am interested to see what a (2?-)flat 2-connection on a 2-covering space does, if we think of it as an assortment (well, perhaps two) of differential forms on the Lie groupoid $X^{(2)}$, with values in some sort of Lie 2-group (say the [[string 2-group]], I'm not sure what other ones can bring to the table). Another thing that might be done is given an integral 3-form (or more generally a 3-class in differential cohomology) on a manifold, we can pull it back to the 2-connected cover, and see what geometric constructions can be extracted, and when they descend to the manifold. The precedent for thinking this sort of thing is constructions that need 1-connectedness, and so pass to the universal covering space.

   (I’m going to move this into the atrium as this isn’t about nlab pages anymore)

   thanks, I probably needed some air anyway :)

   What I’m thinking about is how to construct the darn thing, at least in the smooth case, and have not so clear a picture as to why one might want to! This is presumably all covered in your thesis, which I should now go and read, I guess, but if there’s a quick explanation that you could easily give, then that would help too (think of it as a first draft of the introduction!).

   well, here’s a reeal quick attempt. I should probably expand this into something worthy of a blog post, if the cafe is interested.

   ---

   The universal covering space (of a pointed space/manifold) has as its underlying set the source fibre of the fundamental groupoid. The trick is just to turn it into a space or manifold, and this means you need to make $\Pi_1(X)$ into a topological or Lie groupoid. What’s particularly nice is that the functor $Diff \to Top_{wc} \stackrel{\Pi_1}{\to} Gpd$, for the sort of spaces we are interested in (well-connected spaces), lifts first to $TopGpd$ and then to $LieGpd$ through the obvious forgetful functors. Taking the source fibre of a pointed groupoid is then functorial wrt strictly pointed functors, so the whole construction is functorial.

   It then obvious* that the 2-connected cover of a space/manifold should be the source fibre of a suitably topologised/smoothified fundamental bigroupoid. I did the topological case in my thesis, but there is at least one other functorial construction of a 2-connected cover of a topological space, namely via simplicial sets and the functorial n-truncations available there. Aside from providing concrete examples of non-bundle-gerbe 2-bundles, of which there was a paucity in the literature, the whole construction was meant to be a signpost pointing to how to do the smooth case.

   Now of course the functor $Diff \stackrel{\Pi_2^{Lie}}{\to} Bigpd$ (the 1-category of bigroupoids) doesn’t factor through $Top_{wc} \stackrel{\Pi_2^{top}}{\to} Bigpd$ strictly, but it does up to equivalence. My thesis constructed the lift of the latter functor to $TopBigpd$, but really I want to lift the former to $LieBigpd$. Then for pointed spaces we get strictly pointed 2-functors and so we can describe a functor $X \mapsto X^{(2)}$ associating to a space the 2-connected cover, and using your work, the same for pointed manifolds.

   One potential application is to, as I did in my thesis for the topological case, reverse engineer the definition of smooth 2-covering space (actually this follows almost formally, but it’s a good idea to have an example around to help). Then I am interested to see what a (2?-)flat 2-connection on a 2-covering space does, if we think of it as an assortment (well, perhaps two) of differential forms on the Lie groupoid $X^{(2)}$, with values in some sort of Lie 2-group (say the string 2-group, I’m not sure what other ones can bring to the table). Another thing that might be done is given an integral 3-form (or more generally a 3-class in differential cohomology) on a manifold, we can pull it back to the 2-connected cover, and see what geometric constructions can be extracted, and when they descend to the manifold. The precedent for thinking this sort of thing is constructions that need 1-connectedness, and so pass to the universal covering space.