fibrations as ...

[jim stasheff <jds@math.upenn.edu>]

John and anyone else who cares to weigh in,
here are some comments from the purely topological
or rather homtopy theory side:

For both bundles and fibrations (e.g. over a paracompact base), your
last slogan is the oldest:

  FIBRATIONS WITH FIBER F OVER THE SPACE B
                             ARE "THE SAME" AS
                MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F)

`the same as' referring to homotopy classes

For bundles, Aut(F) now means the structure group of the bundle

This standard approach is great - I'm not knocking it - but Schreier theory
is more general: it's really a branch of "nonabelian cohomology theory".
It's not all that hard to explain, either.  So, I'll explain it and then
talk about various simplifying assumptions people make.

The goal of Schreier theory is to classify short exact sequences of groups:

1 -> F -> E -> B -> 1

for a given choice of F and B.  "Exact" means that the arrows stand
for homomorphisms and the image of each arrow is the kernel of the next.
Here this just means that F is a normal subgroup of E and B is the quotient
group E/F.  Such a short exact sequence is also called an "extension of B
by F", since E is bigger than B and contains F.  The simplest choice is to
let E be the direct sum of F and B.  Usually there are other more
interesting
extensions as well.

To classify these, the trick is to use the analogy between group theory
and topology.

As I explained in "week213", you can think of a group as a watered-down
version of a connected space with a chosen point.  The reason is that
given such a space, we get a group consisting of homotopy classes of
loops based at the chosen point.  This is called the "fundamental group"
of our space.  There's a lot more information in our space than this group.
But pretty much anything you can do for groups, you can do for such spaces.
It's usually harder, but it's completely analogous!

In particular, classifying short exact sequences is a lot like
classifying "fibrations":

1 -> F -> E -> B -> 1

where now the letters stand for connected spaces with a chosen point, and
the arrows stand for continuous maps.   If you're a physicist or geometer
you may prefer fiber bundles to "fibrations" - but luckily, they're so
similar we can ignore the difference in a vague discussion like this.
The idea is basically just that E maps onto B, and sitting over each point
of B we have a copy of F.  We call B the "base space", E the "total space"
and F the "fiber".

If we want to classify such fibrations we can consider carrying the fiber
F around a loop in B and see how it twists around.  For example, if all our
spaces are smooth manifolds, we can pick a connection on the total space
E and see what parallel transport around a loop in the base space B does
to points in the fiber F.  This gives a kind of homomorphism

Omega(B) -> Aut(F)

sending loops in B to invertible maps from F to itself.  And, the cool
thing is: this homomorphism lets us classify the fibration!

Here I say "kind of homomorphism" since Omega(B), the space of loops in B
based at the chosen point, is only "kind of" a topological group: the
group laws only hold up to homotopy.

As Peter May points out, you can use the strict monoid of Moore loops
instead of Poincare's, but kind of homomorphism" still does not mean strict
but rather A-\infty map even though both domain and range are strictly
associative

So intrepreted, the equivalence with B --> Aut F is clear.

If you wnat a strict homomorphism, try using a classical connection -
at least in the smooth case.

 But let's not worry about this
technicality  - especially since I'm being vague about all sorts of other
equally important issues!

The reason I can get away with not worrying about these issues is that
I'm trying to explain a very robust powerful principle - one that can
easily survive a dose of vagueness that would kill a lesser idea.  Namely,
if B is a connected space with a chosen basepoint,

               FIBRATIONS OVER THE BASE SPACE B WITH FIBER F
                          ARE "THE SAME" AS
         HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F.

This could be called "the basic principle of Galois theory", for reasons
explained in "week213".  There I explained the special case where the
fiber is discrete.  Then our fibration called a "covering space", and
the basic principle of Galois theory boils down to this:

                    COVERING SPACES OVER B WITH FIBER F
                           ARE "THE SAME" AS
    HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F.

Okay.  Now let's use the same principle to classify extensions of a group
B by a group F:

1 -> F -> E -> B -> 1

The group B here acts like "loops in the base".  But what acts like
"automorphisms of the fiber"?

You might guess it's the group of automorphisms of F.  But, it's
actually the *2-group* of automorphisms of F!

A 2-group is a categorified version of a group where all the usual group
laws hold up to natural isomorphism.  They play a role in higher gauge
theory like that of groups in ordinary gauge theory.  In higher gauge
theory, parallel transport along a path is described by an *object* in
a 2-group, while parallel transport along a path-of-paths is described
by a *morphism*.  In 2-form electromagnetism we use a very simple "abelian"
2-group, which has one object and either the real line or the circle as
morphism.  But there are other more interesting "nonabelian" examples.

If you want to learn more about 2-form electromagnetism from this
perspective, try "week210".  For 2-groups in general, try this paper:

9) John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-groups,
Theory and Applications of Categories 12 (2004), 423-491. Available
online at
http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html or as math.QA/0307200.

Anyway: it turns out that any group F gives a 2-group AUT(F) where the
objects are automorphisms of F and the morphisms are "conjugations" -

and conjugations correspond to homtopies at the classifying space level

elements of F acting to conjugate one automorphism and yield another.
And, extensions

1 -> F -> E -> B -> 1

are classified by homomorphisms

B -> AUT(F)

where we think of B as a 2-group with only identity morphisms.  More
precisely:

                EXTENSIONS OF THE GROUP B BY THE GROUP F
                            ARE "THE SAME" AS
               HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F)

It's fun to work out the details, but it's probably not a good use of
our time together grinding through them here.   So, I'll just sketch
how it works.

Starting with our extension

         i     p
1 --> F --> E --> B --> 1

we pick a "section"

                s
            E <-- B

meaning a function with

p(s(b)) = b

for all b in B.  We can find a section because p is onto.  However,
the section usually *isn't* a homomorphism.

Given the section s, we get a function

alpha: B -> Aut(F)

where Aut(F) is the group of automorphisms of F.  Here's how:

alpha(b) f = s(b) f s(b)^{-1}

However, usually alpha *isn't* a homomorphism.

So far this seems a bit sad: functions between groups want to be
homomorphisms.  But, we can measure how much s fails to be a homomorphism
using the function

g: B2 -> F

given by

g(b,b') = s(bb') s(b')^{-1} s(b)^{-1}

Note that g = 1 iff s is a homomorphism.

We can then use this function g to save alpha.  The sad thing about
alpha is that it's not a homomorphism... but the good thing is, it's
a homomorphism up to conjugation by g!  In other words:

alpha(bb') f = g(b,b') [alpha(b) alpha(b') f] g(b,b')^{-1}

Taken together, alpha and g satisfy some equations ("cocycle conditions")
which say precisely that they form a homomorphism from B to the 2-group
AUT(F).  Conversely, any such homomorphism gives an extension of B by F.

In fact, isomorphism classes of extensions of B by F correspond
in a 1-1 way with isomorphism classes of homorphisms from B to AUT(F).
So, we've classified these extensions!

In fact, something even better is true!  It's evil to "decategorify" by
taking isomorphism classes as we did in the previous paragraph.  To avoid
this, we can form a groupoid whose objects are extensions of B by F, and a
groupoid whose objects are homomorphisms B -> AUT(F).  I'm pretty sure
that if you form these groupoids in the obvious way, they're equivalent.
And that's what this slogan really means:

                EXTENSIONS OF THE GROUP B BY THE GROUP F
                            ARE "THE SAME" AS
               HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F)

Ah, your 2-group must be hiding my homotopies - I'd love to see that

Next, let me say how Schreier theory reduces to more familiar ideas in two
special cases.

People have thought a lot about the special case where F is abelian and
lies in the center of E.  These are called "central extensions".  This
is just the case where alpha = 1.  The set of isomorphism classes of
central extensions is called H2(B,F) - the "second cohomology" of B
with coefficients in F.

People have also thought about "abelian extensions".  That's an even
more special case where all three groups are abelian.  The set of
isomorphism classes of such extensions is called Ext(B,F).

Since we don't make any simplifying assumptions like this in Schreier
theory, it's part of a subject called "nonabelian cohomology".  It was
actually worked out by Dedecker in the 1960's, based on much earlier
work by Schreier:

10) O. Schreier, Ueber die Erweiterung von Gruppen I, Monatschefte fur
Mathematik and Physick 34 (1926), 165-180.  Ueber die Erweiterung von
Gruppen II, Abh. Math. Sem. Hamburg 4 (1926), 321-346.

11) P. Dedecker, Les foncteuers Ext_Pi, H2_Pi and H2_Pi non abeliens,
C. R. Acad. Sci. Paris 258 (1964), 4891-4895.

More recently, Schreier theory was pushed one step up the categorical
ladder by Larry Breen.  As far as I can tell, he essentially classified
the extensions of a 2-group B by a 2-group F in terms of homomorphisms
B -> AUT(F), where AUT(F) is the *3-group* of automorphisms of F:

12) Lawrence Breen, Theorie de Schreier superieure, Ann. Sci. Ecole Norm.
Sup. 25 (1992), 465-514.  Also available at
http://www.numdam.org/numdam-bin/feuilleter?id=ASENS_1992_4_25_5

We can keep pushing Schreier theory upwards like this, but we can also
expand it "sideways" by replacing groups with groupoids.  You should have
been annoyed by how I kept assuming my topological spaces were connected
and equipped with a specified point.   I did this to make them analogous
to groups.  For example, it's only spaces like this for which the
fundamental
group is sufficiently powerful to classify covering spaces.  For more
general
spaces, we should use the fundamental *groupoid* instead of the fundamental
group.  And, we can set up a Schreier theory for extensions of groupoids:

13) V. Blanco, M. Bullejos and E. Faro, Categorical non abelian cohomology,
and the Schreier theory of groupoids, available as math.CT/0410202.

In fact, these authors note that Grothendieck did something similar
back in 1971: he classified *all* groupoids fibered over a groupoid
B in terms of weak 2-functors from B to Gpd, which is the 2-groupoid of
groupoids!  The point here is that Gpd contains AUT(F) for any fixed
groupoid F:

14) Alexander Grothendieck, Categories fibrees et descente (SGA I),
Lecture Notes in Mathematics 224, Springer, Berlin, 1971.

Having extended the idea "sideways" like this, one can then continue
marching "upwards".  I don't know how much work has been done on this,
but the slogan should be something like this:

                  n-GROUPOIDS FIBERED OVER AN n-GROUPOID B
                             ARE "THE SAME" AS
            WEAK (n+1)-FUNCTORS FROM B TO THE (n+1)-GROUPOID nGpd

aha, weak or lax as much above will likely be also

Grothendieck also studied this kind of thing with categories replacing
groupoids, so there should also be an n-category version, I think...
but it's more delicate to define "fibrations" for categories than
for groupoids, so I'm a bit scared to state a slogan suitable for
n-categories.

However, I'm not scared to go from n-groupoids to omega-groupoids, which
are basically the same as spaces.  In terms of spaces, the slogan goes
like this:

                        SPACES FIBERED OVER THE SPACE B
                               ARE "THE SAME" AS
                    MAPS FROM B TO THE SPACE OF ALL SPACES

This is how James Dolan taught it to me.  Most mortals are scared of "the
space of all spaces" - both for fear of Russell's paradox, and because we
really need a *space* of all spaces, not just a mere set of them.  To avoid
these terrors, you can water down Jim's slogan by choosing a specific space
F to be the fiber:

                  FIBRATIONS WITH FIBER F OVER THE SPACE B
                             ARE "THE SAME" AS
                MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F)

where AUT(F) is the topological group of homotopy self-equivalences of F.
The fearsome "space of all spaces" is then the disjoint union of the
classifying spaces of all these topological groups AUT(F).  It's too large
to be a space unless you pass to a larger universe of sets, but otherwise
it's perfectly fine.  Grothendieck invented the notion of a "Grothendieck
universe" for precisely this purpose:

14) Wikipedia, Grothendieck universe,
http://en.wikipedia.org/wiki/Grothendieck_universe

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