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January 31, 2008
Geometric Representation Theory (Lecture 24)
Posted by John Baez
This time in the Geometric Representation Theory Seminar, I finished my lightning review of the quantum harmonic oscillator. Then I moved on to a lightning review of how to groupoidify it!
I’ve already explained this stuff in vastly greater detail back in the Fall 2003, Winter 2004 and Spring 2004 sessions of the seminar — you can see extensive notes by clicking on the links. This time we’re whizzing through this material very fast. Then we’ll use it to groupoidify a bunch of representations of the Lie algebras . Then we’ll try to -deform the whole story! At that point, we’ll hook up with what Jim has been explaining about quiver representations and quantum groups.
January 30, 2008
L-infinity Associated Bundles, Sections and Covariant Derivatives
Posted by Urs Schreiber
Here is the alpha -version of a plugin for the article -connections (pdf, blog, arXiv) which extends the functionality of the latter from principal -connections to associated -connections:
Sections and covariant derivatives of -algebra connections (pdf, 8 pages)
Abstract. For every -algebra there is a notion of -bundles with connection, according to [SSS]. Here I discuss how to describe
- associated -bundles;
- their spaces of sections;
- and the corresponding covariant derivatives
in this context.
Introduction. Representations of -groups are usually thought of as -functors from the -group into the -category of representing objects. In the program [BaezDolanTrimble] one sees that possibly a more fundamental perspective on representations is in terms of the corresponding action groupoids sitting over the given group.
This is the perspective I will adopt here and find to be fruitful.
The definition of -modules which I proposed in -modules and the BV-complex (pdf, blog) can be seen to actually comply with this perspective. Here I further develop this by showing that this perspective also helps to understand associated -connections, their sections and covariant derivatives.
January 29, 2008
Geometric Representation Theory (Lecture 23)
Posted by John Baez
In this session of the Geometric Representation Theory Seminar, I was stuck in the local courthouse on jury duty. (I wasn’t selected to be a juror.) So, Jim Dolan continued his story from last time: groupoidifying the Hall algebra of a quiver.
The ultimate goal is to categorify the theory of quantum groups.
January 28, 2008
Differential Forms and Smooth Spaces
Posted by Urs Schreiber
As we have discussed at length in the entry on Transgression, differential graded commutative algebras (DGCAs for short) have a useful relation to presheaves on open subsets of Euclidean spaces (smooth spaces, for short):
- every smooth space has a DGCA of differential forms on it;
- and every DGCA sits inside the algebra of differential forms of some smooth space .
On top of that, every finite dimensional DGCA can be regarded as the Chevalley-Eilenberg algebra of some Lie -algebroid , which linearizes some Lie -groupoid.
Here I want to talk about my expectation that
The smooth space associated to any Lie -algebroid this way plays the role of the space # of the Lie -groupoid integrating .
As motivation and plausibility consideration, recall that in rational homotopy theory and notably in the work of Getzler and Henriques, one obtains a simplicial space from by defining its collection of -simplices to be the collection of -valued forms on the standard -simplex…
… and notice that with the above and using the Yoneda lemma, we can equivalently think of this as the collection of -simplices in :
Mapping simplices into a smooth space is like computing its fundamental -groupoid , thought of as a Kan-complex. In simple situations, notably when is an ordinary Lie algebra, also the ordinary fundamental (1-)groupoid is of interest. And I think in this case, where the right hand side simply denotes the one-object groupoid with as its space of morphisms.
I am thinking, that hitting everything you see in sections 6 onwards in Lie -connections (blog, pdf, arXiv) with should have various nice consequences.
I want to better understand how nice exactly. That involves better understanding the properties of these functors in light of the above expectation.
All help is very much appreciated.
January 25, 2008
The Yoneda Embedding as a Reflection
Posted by John Baez
Guest post by Mike Stay
In my last post I explained how the Yoneda embedding was secretly the same as the ‘continuation passing transform’. Here’s a nice pictorial way to think about it.
January 24, 2008
Integration Without Integration
Posted by Urs Schreiber
In some comments to On Lie -tegration and Rational Homotopy Theory, starting with this one, I began thinking about defining integration of forms over a manifold in terms of a mere passage to equivalence classes.
There is a big motivation here coming from the observation in Transgression of -Transport and -Connection, that fiber integration is automatically induced by hitting transport functors with inner homs.
We want the Lie -algebraic version of this, in order to possibly understand how to perform the path integral of a charged -particle coupled to a Lie -algebraic connection as in the last section of -connections and applications to String- and Chern-Simons -transport (arXiv:0801.3480).
I think I made some progress with understanding this in more detail. I talk about that here:
Integration without Integration (pdf, 6 pages)
Abstract: On how transgression and integration of forms comes from internal homs applied on transport -functors, on what that looks like after passing to a Lie -algebraic description and how it realizes the notion of integration without integration.
Classifying Spaces for 2-Groups
Posted by John Baez
These days I’ve been working hard finishing off papers. Now you can see this one on the arXiv:
- John Baez and Danny Stevenson, The classifying space of a topological 2-group.
Or, you can just read this summary…
January 18, 2008
Geometric Representation Theory (Lecture 22)
Posted by John Baez
This time in the Geometric Representation Theory Seminar, Jim introduces a new example: the Hall algebra of a quiver.
January 17, 2008
Spaceoids
Posted by Urs Schreiber
guest post by Paolo Bertozzini – a pdf version of this post is here
In the discussion following the posts
“The Principle of General Tovariance” and
“Australian Category Theory”,
Urs Schreiber, Kea and David Corfield have been mentioning the research work on “categorical non-commutative geometry” that I am carrying on with my collaborators Roberto Conti (now in University of Newcastle - Australia) and Wicharn Lewkeeratiyutkul (in Chulalongkorn University - Bangkok). It is a pleasure to reply with some more detailed information on some of these topics.
Specifically this post is mainly concerned with the “horizontal categorification” (or “oidization/many-objectification” as John Baez prefers to call it) of the notion of (compact Hausdorff topological) space.
Let us start with some simple but intriguing questions:
- What might be a good categorical version of the notion of space?
- Might non-commutative geometry provide some guidance towards at least one of the possible answers to the previous question?
101 things to do with a 2-classifier
Posted by David Corfield
Thanks to Tom and Todd, I have an answer to the problem I posed of what does the classifying for 2-categories.
- At level 0, we have the set inclusion .
- At level 1, we have the forgetful functor .
- At level 2, we have the forgetful 2-functor .
is what I’m calling the 2-category of pointed categories , but where a map is a functor together with a map in .
If there’s a name already for this 2-category, do please let me know.
January 16, 2008
Slides: On the BV-Formalism (BV Part XI)
Posted by Urs Schreiber
In the process of wrapping up what has happened so far (part I, II, III, IV, V, VI, VII, VIII, IX, X) I am working on this set of pdf-slides (should be printable, no fancy overlay tricks this time; if you read it online, navigate like a web-site (use your pdf-reader’s back-button!))
Abstract. We try to understand the Batalin-Vilkovisky complex for handling perturbative quantum field theory. I emphasize a Lie -algebraic perspective based on [Roberts-S., Sati-S.-Stasheff] over the popular supergeometry perspective and try to show how that is useful. A couple of ex