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Max New: at (129.10.9.38) has put a query on the generalized universal bundle entry. It says:
I don’t understand the above diagram, what is the point $* \to X$ in question? and how does this relate to the universal bundle? In particular, there is a sequence below that has a map from $\Omega_{pt} \to \mathbf{E}_{pt}$ but I don’t see how to construct that from the above.
So I think this is a typo, but I don’t know enough to correct it.
Hi, this was me (and my name really is Max New).
After looking at the loop space object page more closely I’m pretty sure $X$ should be $B$, $g$ should be the identity and then $P$ would be $E_{pt}$, though the text is also not explicit about the fact that $E_{pt}$ can be constructed as a comma object, so I’d like confirmation before changing it.
Max, I carefully did not assume your title was a pseudonym. (The lowercase n was a typo, sorry). The ‘named’ was to allow for the possibility that it was! I have amended the wording.
Welcome to the forum.
My point in copying your query here was to draw the attention of others to your query. (Putting a query on a page is not that good a way of asking it as some pages are rarely visited.)
On the other hand, the nForum is a good way to ask questions about nLab articles. :-)
Welcome to the forum, Max!
Regarding the actual question, a couple of paragraphs above the text defined “a $B$-bundle on some object $X$”, so I expect the intent of the author is that we are still in that situation; $g$ is the classifying morphism of the $B$-bundle $P\to X$. The point $\ast\to X$ is arbitary; what the text would then be trying to say is that the fiber of $P\to X$ over any point $\ast \to X$ is $\Omega_{pt} B$, although if this is the intent then the surrounding text needs clarification. In particular I think the left-hand square in the diagram should be a strict pullback, hence have no $\Downarrow$ in it; the right one would be a strict pullback if it had $E_{pt} \to B$ on the right rather than $\ast \to B$. In the special case when $X$ is $B$ and $g$ is the identity, then $P = E_{pt}$, and we see that the fiber of the generalized universal bundle itself over the basepoint is the loop space $\Omega_{pt} B$. Perhaps it would be clearer to state only that case, as you suggest. Anyone else have an opinion?
Mike, I don’t see how the same $\Omega_{pt} B$ could arise from an arbitrary pointed $X$. For instance, what if $X = *$ and $g$ picks out a point that has no paths from pt? Then $P$ would be empty and so would $\Omega_{pt}$, but I think $\Omega_{pt}$ should be all the endomorphisms of pt.
I would definitely state the basic case, where one just takes the fibre using the same point, on its own first, straight after the first definition in the section (i.e. before the $X$ enters).
I also agree with Max: the claim is wrong as stated. What one gets in general is the ’object of paths’ from one of the points to the other. This is certainly not isomorphic or even homotopy equivalent (where this is defined using the interval object) to the loop object in general, as his example shows.
I would say that the use of both ’lax pullback’ and ’comma object’ is also not optimal. I would be fine with ’homotopy pullback’, but best would probably be ’double mapping co-cylinder’.
You’re right, probably whoever wrote it was assuming that $B$ was connected. And I also would prefer “homotopy pullback” – “lax pullback” and “comma object” sound too directed.
Yes, if there is a path (with respect to the given interval object) between the two points (being careful about direction if the interval object does not an involution structure), then one will be able to use the universal property of the double mapping co-cylinder to construct a homotopy equivalence between the loop object and the object of paths.
It seems that the original author did intend for directed-ness to be considered. However, I still think that ’homotopy pullback’ stays closer to the motivating examples, whilst not disallowing being understood to mean ’directed homotopy pullback’. Because probably, if one really wishes to keep the directed-ness in mind, then one should speak of a directed loop space object, etc: one has to be careful about this everywhere.
I don’t really think it makes sense to use any of this language in the directed case.
Yes, I agree really. In particular, thinking about it fractionally more, I think one needs the involution structure to get both directions of the homotopy equivalence I was referring to in the first paragraph of #9.
It looks like Urs Schreiber added this: https://ncatlab.org/nlab/revision/diff/generalized+universal+bundle/12, is there an easy way to ping him to this thread (besides just e-mail of course)?
I don’t know of any ping mechanism, but my expectation is that Urs reads every nForum post. I’d email him if he doesn’t reply in a day or two.
Or at least give him the weekend off.
Indeed, David. :-)
It would be more in the wiki spirit to just fix it ourselves; Urs is plenty busy already.
Ok, I’ve updated it based on my understanding. I’d like to argue against making the section more homotopy-y because the page is explicitly meant to also describe various higher Grothendieck constructions, and I was using it to get a handle on the correspondence between functors into a category and normal lax functors to Dist, so the more categorical terminology was quite helpful.
Looks fine to me, thanks!
This is a relatively ancient entry (I have now fixed formatting of the diagrams), started before much higher categorical sophistication was around here.
Your comment seems to pertain to this section. This would certainly deserve to be expanded. If you could add your discussion there, it would be a nice contribution.
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