Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
started an entry associated infinity-bundle
in order to summarize the thesis by Matthias Wendt on associated $\infty$-bundles in arbitrary $(\infty,1)$-toposes, generalizing the classical old results by Stasheff and May from $\infty Grpd$.
Also added some remarks on the relation to the discussion at principal infinity-bundle. Hopefully to be continued tomorrow.
I have added a literature section to fiber bundle (not infinity version, sorry).
Hi Jim!
cf. also representations up to homotopy do we have an entry for that?
Yes, a discussion is at infinity-representation .
is there somewhere buried in there a comparison with actions up to strong homtopy?
Yes, see at higher parallel transport the sections whose title begins with “Flat $\infty$-parallel transport…”.
Mike, here a message to you (or, of course, anyone else who has looked into these questions):
I am writing up a purely $\infty$-topos theoretic proof (meaning: no presentations, essentially a pseudo-formal HoTT proof except that it invokes groupoid objects and their realization) of the following (non-surprising) fact:
Fact In an $\infty$-topos $\mathbf{H}$, for $V$ any object, $V$-fiber $\infty$-bundles over some $X$ are classified by $H^1(X, \mathbf{Aut}(V)) = \pi_0 \mathbf{H}(X, \mathbf{B}\mathbf{Aut}(V))$, this being induced by the map that sends an $\mathbf{Aut}(V)$-principal $\infty$-bundle to its canonically associated $V$-bundle.
Non-surprising as this may be, it directly implies the results referenced in the References at associated infinity-bundle as a special case (the 1-localic case) while itself using only a few lines of abstract nonsense. And the existence of object classifiers / univalence, of course.
While these kinds of general statements are not widely known, I can imagine that they are very clear to a handful of experts. Such as you. On the other hand, I am not aware that these kinds of statements are in the literature.
Or are they? Since we see here a vast overlap of bundle theory that was traditionally rooted in topology and differential geometry with structures now considered in type theory, it may be that in slight disguise this appears elsewhere.
Well, the consideration of these things in general $(\infty,1)$-toposes is not really traditional. (-: As you probably know, in the case $\mathbf{H}=\infty Gpd$, this is essentially the central theorem of “Classifying Spaces and Fibrations”. I believe the only other $(\infty,1)$-toposes that belong to “classical algebraic topology” are slices of $\infty Gpd$ (parametrized homotopy theory) and diagrams on orbit categories (equivariant homotopy theory). People have certainly studied bundles and classifying spaces in those contexts, but I don’t know whether theorems of this generality are classical in those cases.
Thanks, Mike, for the reaction.
Yes, I am well aware that these considerations are not traditional. I just thought I’d check to which degree you, for instance, might find these kinds of statements by now to be an “obvious” consequence of $\infty$-topos axioms, or whether you have seen such statements claimed publically.
As you probably know, in the case H=∞Gpd, this is essentially the central theorem of “Classifying Spaces and Fibrations”.
Yes. And as I said: in fact all the References that are cited in the entry are direct special cases of this general statement, including the one in the article by Wendt, on simplicial presheaves over a 1-site.
and diagrams on orbit categories (equivariant homotopy theory).
Ah, that’s a good point. I haven’t looked into these for a while. I should check the literature again to compare. Can you point me somewhere specifically for these kinds of classifying statements (don’t worry if not, I can dig it out myself, but if you know it off the top of your head, you could safe me some searching).
I don’t know where to look off the top of my head. I’m certainly not surprised that this is true in an $(\infty,1)$-topos, and I can guess how to go about proving it using quotients of groupoid objects; it seems like it ought to be fairly straightforward. I don’t think I’ve heard it stated explicitly anywhere, but I’m not an expert on the $(\infty,1)$-topos literature.
I have expanded the References at associated infinity-bundle a bit further, prompted by seeing the new article On the classification of fibrations.
Since this caused in me the feeling that possibly some groups working on this are not aware of what happens in other corners, I added to the References-section also
a pointer to the References at univalence
a pointer to the subsection at object classifier which discusses how the classical result follows from the existence of the object classifie in $\infty Grpd$.
Finally I have allowed myself to also mention our writeup.
I will now carry part of this discussion about references to a reply to Jim Stasheff over on the $n$Café here.
1 to 10 of 10