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• Something on definite description.

• this table used to be hidden at supersymmetry, but it really ought to cross-link its entries. Therefore here its stand-alone version, for !inclusion

• changed page name to make it fit more systematic naming pattern

• I am moving the following old query box exchange from orbifold to here.

old query box discussion:

I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely does not view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.

Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?

end of old query box discussion

• added to groupoid a section on the description in terms of 2-coskeletal Kan complexes.

• Added redirect for missing link at Banach algebra section “2. Examples”.

Anonymous

• added the statement (here) that of all finite subgroups of $SU(2)$, $Q_8$ is a proper subgroup of the three exceptional ones.

Checking normality of this subgroup, I noticed that there is an issue with another item of the entry here, where it used to claim that a finite group is Hamiltonian precisely of it “contains a copy of $Q_8$”. But this can’t be, can it. I changed it to saying that every Hamiltonian group contains $Q_8$ as a subgroup, which I suppose is what was meant.

[edit: I see now that the statement that I changed back to was made already by Thomas Holder in rev 3, while the statement I removed was made by Thomas in rev 4. Thomas, if you see this, please let me know. ]

• Page created, but author did not leave any comments.

• 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”

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:

• it is the classifying functor $F : C \to Cat$ which should be addressed as the “(parallel) transport”, while the corresponding fibration is its “action object” as in action groupoid, i.e. the thing whose objects are all possible things that the parallel transport can transport and whose morphisms take these things to the image of that transport. So it’s a subtle difference, but an important one.

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

• started a bare minimum at Poisson-Lie T-duality, for the moment just so as to have a place to record the two original references

• added to coalgebra for an endofunctor the example of the real line as the terminal coalgebra for some endofunctor on Posets.

There are more such characterizations of the real line, and similar. I can't dig them out right now as I am on a shky connection. But maybe somebody else can. Or I'll do it later.

• Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the permutation representation in the standard representation theory books like in

• Claudio Procesi, Lie groups, an approach through invariants and representations, Universitext, Springer 2006, gBooks

I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.

Edit: new (related) entries for Claudio Procesi and Arun Ram.

• Created page, mainly to record a bunch of references that I am trying to collect. Additional suggested references would be welcome!

• trying to collect references on the state-of-the-art of computer simiulations on cosmic structure formations. Will try to expand as I find more…

• The old webpage link seemed to be dead, so I have replaced it with both the HCM and the HIM links in Bonn.

• added articles under “Selected writings”

• to be !include-ed as a floating table of contents into relevant entries

• brief category:people-entry for hyperlinking references at non-abelian T-duality and elsewhere

• brief category:people-entry for hyperlinking references at non-abelian T-duality and elsewhere

• added to Eckmann-Hilton argument the formal proposition formulated in any 2-category.

BTW, doesn’t anyone have a gif with the nice picture proof?

• Add missing nullary condition; note unbiased version.