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I have edited fibration
promted by an email question I have added more information on when the pullback of a fibration is a homotopy pullback;
in the discussion of “transport” in topological spaces I added a pointer to Flat ∞-parallel transport in Top which gives details;
I fixed a mistake where quasifibration was mentioned and pointed to, but, fibration in the Joyal model structure was meant (despite the previous warning of exactly this trap…)
I added a subsection “Related concepts”
I added loads and loads of hyperlinks to the keywords.
FInally, I noticed that the following old discussion was sitting there, which hereby I move fromthere to here
begin forwarded discussion
+–{.query} Tim: I do not quite agree with ’transport’ as being the main point of fibrations. Rather ’lifting’ is the main point, in particular lifting of homotopies, at least in topological situations. For transport, one needs connections of some sort to get things working well, but in many cases there is only a very weak notion of action, so perhaps that should be derived as a property rather than taken as a ’defining property’ in some sense.
Perhaps a reference to Stasheff and Wirth
James Wirth & Jim Stasheff
Homotopy Transition Cocycles
math.AT/0609220.
and the discussion
http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html
on the cafe would be a good idea to add.
Urs: In situations where one wants to talk of transport, the fibration usually arises as the pullback of some “universal fibration”, a generalized universal bundle. For instance (split op-)fibrations of categories are precisely the pullbacks of the universal $Cat$-bundle $Cat_* \to Cat$ along a functor $F : C \to Cat$.
If one looks at this kind of situation where we do have an established notion of (parallel) transport one sees:
For instance, to make this more concrete, consider the category of smooth groupoids (which is a category of fibrant objects), let for any manifold $X$ the groupoid $P_1(X)$ be the groupoid of smooth thin-homotopy classes of paths in $X$, let $G$ be any Lie group, $\mathbf{B} G$ the corresponding one-object Lie groupoid and consider the _universal fibration _ $\mathbf{E} G \to \mathbf{B}G$ – the groupoid incarnation of the universal $G$-bundle as described at generalized universal bundle. Then
Theorem: $G$-bundles with connection on $X$ are equivalent to functors $tra : \widehat{P_1(X)} \to \mathbf{B}G$ out of acyclic fibrations $\widehat{P_1(X)} \to P_1(X)$ over $P_1(X)$ (i.e. smooth anafunctors $P_1(X) \to \mathbf{B}G$). These functors are literally the corresponding “parallel transport”: indeed, evaluated on a path $\gamma$ in $X$ there is locally a 1-form $A \in \Omega^1(X, Lie(G))$ such that the group element $tra(\gamma)$ is the traditional parallel transport of that 1-form, $tra(\gamma) = P exp(\int_\gamma A)$.
Now, we can form the fibration which is associated with this parallel transport, namely the pullback
$\array{ tra^* \mathbf{E} G &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ \widehat{P_2(X)} &\stackrel{tra}{\to}& \mathbf{B}G \\ \downarrow \\ P_2(X) } \,.$This fibration $tra^* \mathbf{E}G \to \widehat{P_2(X)}$ is what is properly speaking the action groupoid of $tra$ acting on the fibers of the principal $G$-bundle.
Mike: Can you clarify the distinction between “lifting” and “transport”? In what way does the lifting of a path $f$ starting at a point $e$ not transport $e$ along $f$? Certainly in geometric situations to get a parallel notion of transport, you need a connection, but I see that as a stronger requirement.
forwarded discussion continued in next entry
continuation of forwarded discussion
Mike adds: In fact, any topological fibration over $X$ (not just a bundle) induces a “transport” map from (the fundamental $\infty$-groupoid of) $X$ to the $(\infty,1)$-category $Top$, taking each point to its fiber. Putting extra structure on the fibration, such as making it a $G$-bundle, corresponds to restricting the codomain of the transport functor to land in some smaller subcategory, such as $\mathbf{B}G$. A reference in classical homotopy theory is May’s “Classifying spaces and fibrations.”
I agree with what Urs said that the “transport” properly refers to the corresponding functor into $Top$, $Cat$, or $\mathbf{B}G$, with the fibration being the “action object.” Then the property of “being a fibration” means “can be equipped with a transport” or “can be obtained from a transport.” These are all important points to be added, but I think that what I wrote is still correct.
-Tim The transport aspect of course is there, but as the lifting is not unique the idea of the transport being ’up to coherent homotopy’ or ’$\infty$-homotopy is much more sophisticated than the simple point of the existence of a ’lifting’. My point is that although correct your definition may be accused of being ’mathematics made difficult’. To have a definition of fibration that somehow requires one to have understood $(\infty,1)$-categories and the fundamental $\infty$-groupoid of a space is, to me, the wrong way around. The simply stated lifting property leads eventually naturally to the beautiful transport description (and is more general, more classically based, and probably more approachable). That process is only recently understood and is still only understood by a handful of people who work with fibrations.
If you look at the classical treatment of fibre spaces (and I found a copy of Grothendieck’s 1950s notes on them in a pile of stuff today), there is no thought of transport, just of lifting. Serre, who revolutionised the algebraic topology of fibre spaces and fibrations is using lifting, not transport. The link between the transport functor and the notion of fibration emerged via the ’Grothendieck construction’ (due to Ehresmann as well.. and there transport was part of the story that he wanted) but was part of the image of fibre spaces\fibrations as being like semi-direct products and that took a long time to get to the present state.
Mike: Well, I think a lot of what we do here could be accused of being “mathematics made difficult.” Cf. the first sentence of group. But I believe that it is useful and important to have a correct conceptual understanding, in addition to any technically simpler definition that one works with in practice.
For instance, I believe that the “real” definition of a monoidal category requires that “all diagrams of constraints commute,” while the fact that it’s enough to check two particular diagrams is a happy accident. Presenting the pentagon and unit axioms as the God-given definition of a monoidal category, without mentioning that the only reason this is an okay definition is because you can prove a coherence theorem from it, leads the student quite naturally to wonder why God likes pentagons.
Likewise, I believe that the only reason the definition of a fibration via lifting properties is a correct definition is that it does lead to the complete functoriality up to higher homotopies. (One doesn’t need complicated notions of $\infty$-categories to get an intuitive feel for how this works: it’s enough to observe that lifting of 2-cells gives you homotopy-uniqueness for lifts of paths, lifting of 3-cells gives you homotopy-uniqueness for lifts of 2-cells, etc.) The fact that it took many years for this to be understood is not, to me, an argument against its truth, or against its expository helpfulness.
Also, I wrote this page not to be just about the topological notion of fibration, but to include categorical (Grothendieck) fibrations as well and show the commonality between them. And for that I think the notion of transport is indispensable, since categorical fibrations can’t be defined by a simple lifting property.
fowarded discussion to be continued in the next comment
continuation of forwarded discussion
Mike, again: Okay, I basically rewrote the whole page, trying to take your comments into account. What do you think of it now?
Tim : It looks good. My objection was, sort of, pedagogical as I did not want to put off the reader who might be wanting to understand fibrations from the infinity cat point of view. You nicely ease the way in, suggesting how the understanding of the basic classical notion evolves. I like it. Thanks.
PS. Wikipedia entries for fibration and fibred category exist and could be used if you think they are consistent enough with you wishes for this.
Urs: thanks, nice entry. Eventually it would be nice if we could say more about which fibrations are fibrations: which notion of fibred ($n$-)categories/groupoids are fibrations with respect to which model structure. I once tried to collect a bit of data on this, but I need to dig out my old notes, and they were likely very incomplete anyway.
Ronnie: I made a few minor changes, but I also find it odd to define a fibration in terms of a hypothetical fundamental $\infty$-groupoid of $B$. What is this animal?
With regard to transport I like the old papers of Philip R. Heath on operations and fibrations-
Heath, Philip R. Groupoid operations and fibre-homotopy equivalence. I. Math. Z. 130 (1973), 207–233. (the first of 3 papers)
If $p:E \to B$ is a fibration then there is an operation of the paths on the base on the fibres the laws of an operation holding up to homotopy.
At Hull, Phil and I were amused by the fact that the unit interval $I=[0,1]$ is a cogroupoid up to homotopy where homotopy is defined using $I$!. This led to his thesis work on fibrations, getting away from the base point approach of the loops on the base operates on the fibre. Maybe effort should be put in to defining and constructing an $\infty$-cogroupoid (up to homotopy)!
=–
end of forwarded discussion
Jim., this is some very old discussion that I moved over here only because we agree’d to do so, instead of deleting it.
Is there anything in the current entry fibration that you feel needs to be changed? Let’s talk about that.
In section 4, “Fibrations in Category Theory”, Grothendiek and Cartesian fibrations are treated on the same footing saying that the fibration functor $p:E \to B$ produces a pseudofunctor $B \to Cat$, but I think that in the Cartesian case the functors should be $(\infty, 1)$.
I think that whoever wrote this paragraph was using all four terms to mean the same thing. It’s somewhat of an accident of history that our page Grothendieck fibration is about the 2-categorical version and our page Cartesian fibration is about the $(\infty,1)$-categorical one; there’s nothing intrinsically more “Grothendieck” about the 2-categorical version or “Cartesian” about the $(\infty,1)$-categorical one. It might be better if the page currently called “Cartesian fibration” were renamed something like “Cartesian fibration of quasi-categories”. But I agree, because of that accident, the current phrasing on the page fibration is potentially confusing.
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