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September 11, 2006
Wirth and Stasheff on Homotopy Transition Cocycles
In 1965, James Wirth authored a PhD thesis around the description of fibrations
(1) p. E → B p. E to B
when it comes to transition data
(2) p − 1 ( U α ) → t α ← t ¯ α U α × F p ↓ ↓ p U α = U α. array< p^(U_\alpha) \array \\ \stackrel> U_alpha times F p downarrow downarrow p U_alpha = U_alpha > .
Here U α U_alpha are aspects of a great covering of B B by open sets, F F may be the typical fiber. t α t_alpha is really a selected trivialization from the fibration over U α U_alpha. and t ¯ α bar t_alpha its inverse, as much as homotopy.
A brand new arXiv entry now recalls the primary concept of this old operate in modern language:
James Wirth Jim Stasheff
Homotopy Transition Cocycles
The problem looks nearly the same as that familiar in the local trivialization of principal fiber bundles. The important generalization, though, is incorporated in the final clause, which asserts that t α t_alpha and t ¯ α bar t_alpha are just weak inverses of one another.
Because of that, the familiar cocycle equation
for that transition functions
(2) g α β = t ¯ α t β g_ = bar t_alpha t_beta
will generally only support homotopy
If these f ⋯ f_ satisfy a formula postulating a kind of associativity, we obtain a greater cocycle equation as known from gerbes. But, still more generally, this f ⋯ f_cdots itself may be associative only as much as homotopy, and so forth. This yields cocycles from the form because they corespond to two-gerbes, 3-gerbes, etc.
You can nicely summary all of this information when it comes to things i would call a pseudofunctor
(4) U → C. mathbf to C .
where U U is exactly what I understand because the 𐳬h groupoid from the good covering ⊔ α U α sqcup_alpha U_alpha where C C is a few n n -category. Within the topological world this really is known as a functor as much as strong homotopy.
(For many images of how such functors seem like I’m able to point for example for this which. The overall concept of realizing general cocycles as functors from simplices to particular codomains seemed to be formulated by John E. Roberts.)
Wirth labored out how much fibrations are equal to their assortment of transition data (�scent data”).
James F. Wirth
Fiber spaces and also the greater homotopy cocycle relations
PhD thesis, College of Notre Dame, 1965
The tough part would be to rebuild the fibration E E in the transitions between its local trivializations.
For any fiber bundle, we simply go ahead and take space ⊔ α U α × F sqcup_alpha U_alpha times F and identify points that are related under g α β g_. Clearly, for general fibrations this construction is much more involved, because the g α β g_ are not even close to inducing an equivalence relation.
The important tool to make progress may be the mapping cylinder theorem that was mentioned and proven in
James F. Wirth
The mapping cylinder axiom for WCHP fibrations
Pac. J. Math. 54 (2):263-279, 1974 .
I thank Jim Stasheff to make me conscious of the work within the comment portion of this entry. I would possess a comment along with a question, however i will publish those to the comment portion of this entry here.
Published at September 11, 2006 7:15 PM UTC
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Re: Wirth and Stasheff on Homotopy Transition Cocycles
This is the announced comment, in other words some hybrid from a comment along with a question:
John Baez and Candice Bartels have investigated the thought of studying such issues when it comes to n n -bundles .
An n n -bundle is really a (topological or smooth, based on the application) n n -category E E plus a appropriate functor
(1) p. E → B. p. E to B .
where B B is really a discrete n n -category, the bottom space.
When it comes to such n n -bundles there are other of less apparent categorifications of most of the structures familar from ordinary gauge theory, like principal bundles, vector bundles, connections on bundles, etc.
Just like within the theory of homotopy transition cocycles, it’s relatively simple to pass through from the given n n -bundle structure E → B E to B towards the corresponding cocycle data of transitions between local trivializations.
What is not apparent may be the inverse of the construction. I’m not conscious that anybody has attempted to deal with this poor n n -bundles. But possibly a number of Wirth’s work might be applied here.
Personally, I had been already puzzled for a long time in what ought to be the simplest nontrivial example.
Given a Deligne 3-cocycle on the base space B B. i.e. the cocycle describing a U ( 1 ) U(1) -gerbe with connection and curving on B B. I needed to understand which total 2-space E → B E to B with connection
(2) tra. P 2 ( B ) → Trans ( B ) mathrm. P_2(B) to mathrm(B)
has local trivialization data that brings about the required Deligne 3-cocycle.
My method of this was most likely hopelessly unsophisticated when compared with Wirth’s tools. However the solution which i finally created a minimum of includes a form that meets rather nicely the objective of this type of description within the context where I needed to put it on, namely the “physics” of strings in Kalb-Ramond backgrounds.
The answer which i created is described here.
Roughly, the concept is that this:
Towards the U ( 1 ) U(1) -gerbe we might affiliate a PU ( H ) PU(H) principal bundle, and also to that people may affiliate a vector bundle whose fibers are algebras A x A_x of compact operators on some Hilbert space.
Then in the connection and curving data we discover a transport 2-functor, which transmits points x x to algebras A x A_x. pathways x → γ y x stackrel y to some x A_x – A y A_y bimdodules and cobordisms between “parallel” pathways to bimodule homomorphisms.
Now, this yields a complete space which isn’t a category, but simply a collection (with extra structure).
However the 2-category (= bicategory, I usually say n n -category for that weakest possible situation) of bimodules naturally sits within the 2-category Mod Vect mathrm_mathrm of module groups for that monoidal category Vect mathrm .
(3) Bim → Mod Vect mathrm to mathrm_mathrm
an algebra A x A_x is distributed towards the group of created a x A_x -modules (that is itself a module category for Vect mathrm. by tensoring in the right ), a bimodule is distributed to some functor between groups of left modules, and so forth.
By doing this we might consider all of our fibers A x A_x as really denoting a category, namely A x Mod <>_mathrm .
This perspective is great since it manifestly realizes a line bundle gerbe like a 2-vector bundle meaning that objects of Mod Vect mathrm_mathrm can be handled as 2-vector spaces # .
However, it’s not quite as nice, since the total 𔄚-space”
(4) E. = ⊔ x A x Mod E := sqcup_x <>_mathrm
isn’t a topological (significantly less an even) category. A minimum of not without further work and additional assumptions.
Which means this modest observation of mine may be the comment I’ve. The issue obviously is that if maybe Wirth’s result might be useful for understanding such problems with 2-bundles.
Browse the publish Fahrenberg and Raussen on Continuous Pathways
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