The n-Category Café

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 $gl(n)$. Then we’ll try to $q$-deform the whole story! At that point, we’ll hook up with what Jim has been explaining about quiver representations and quantum groups.

Here is the alpha -version of a plugin for the article $L_\infty$-connections (pdf, blog, arXiv) which extends the functionality of the latter from principal $L_\infty$-connections to associated $L_\infty$-connections:

Sections and covariant derivatives of $L_\infty$-algebra connections (pdf, 8 pages)

Abstract. For every $L_\infty$-algebra $g$ there is a notion of $g$-bundles with connection, according to [SSS]. Here I discuss how to describe
$\;\;$ - associated $g$-bundles;
$\;\;$ - their spaces of sections;
$\;\;$ - and the corresponding covariant derivatives
in this context.

Introduction. Representations of $n$-groups are usually thought of as $n$-functors from the $n$-group into the $n$-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 $L_\infty$-modules which I proposed in $L_\infty$-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 $L_\infty$-connections, their sections and covariant derivatives.

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.

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 $X$ has a DGCA $\Omega^\bullet(X)$ of differential forms on it;

- and every DGCA $A$ sits inside the algebra of differential forms of some smooth space $X_A$.

On top of that, every finite dimensional DGCA can be regarded as the Chevalley-Eilenberg algebra $CE(g)$ of some Lie $\infty$-algebroid $g$, which linearizes some Lie $\infty$-groupoid.

Here I want to talk about my expectation that

The smooth space $X_{CE(g)}$ associated to any Lie $\infty$-algebroid $g$ this way plays the role of the space $K(G,n)$ # of the Lie $n$-groupoid $G$ integrating $g$.

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 $S^\bullet_g$ from $g$ by defining its collection of $n$-simplices to be the collection of $g$-valued forms on the standard $n$-simplex…

… and notice that with the above and using the Yoneda lemma, we can equivalently think of this as the collection of $n$-simplices in $X_g$:

$$S^n_g = \mathrm{Hom}_{smooth spaces}( standard n-simplex in \mathbb{R}^n , X_{\mathrm{CE}(g)} ).$$

Mapping simplices into a smooth space is like computing its fundamental $\infty$-groupoid $\Pi_\infty(X_{CE(g)})$, thought of as a Kan-complex. In simple situations, notably when $g$ is an ordinary Lie algebra, also the ordinary fundamental (1-)groupoid $\Pi_1(X_{CE(g)})$ is of interest. And I think $$\Pi_1(X_{CE}(g)) = \mathbf{B}G$$ in this case, where the right hand side simply denotes the one-object groupoid with $G$ as its space of morphisms.

I am thinking, that hitting everything you see in sections 6 onwards in Lie $\infty$-connections (blog, pdf, arXiv) with $g \mapsto X_g \mapsto \Pi_\infty(X_g)$ should have various nice consequences.

I want to better understand how nice exactly. That involves better understanding the properties of these functors $$\array{ DGCAs &&\stackrel{\Omega^\bullet(--)}{\leftarrow} && smooth spaces \\ & \searrow && \swarrow_{\Pi_{\infty}(--)} \\ && infty-groupoids }$$ in light of the above expectation.

All help is very much appreciated.

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.

In some comments to On Lie $N$-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 $n$-Transport and $n$-Connection, that fiber integration is automatically induced by hitting transport functors with inner homs.

We want the Lie $\infty$-algebraic version of this, in order to possibly understand how to perform the path integral of a charged $n$-particle coupled to a Lie $\infty$-algebraic connection as in the last section of $L_\infty$-connections and applications to String- and Chern-Simons $n$-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 $n$-functors, on what that looks like after passing to a Lie $\infty$-algebraic description and how it realizes the notion of integration without integration.

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…

This time in the Geometric Representation Theory Seminar, Jim introduces a new example: the Hall algebra of a quiver.

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?

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 $\{1\} \to \{0, 1\}$.
• At level 1, we have the forgetful functor $(Pointed set) \to Set$.
• At level 2, we have the forgetful 2-functor $(Pointed cat)^+ \to Cat$.

$(Pointed cat)^+$ is what I’m calling the 2-category of pointed categories $(C, c)$, but where a map $(C, c) \to (D, d)$ is a functor $F: C \to D$ together with a map $F(c) \to d$ in $D$.

If there’s a name already for this 2-category, do please let me know.

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!))

On the BV-Formalism

Abstract. We try to understand the Batalin-Vilkovisky complex for handling perturbative quantum field theory. I emphasize a Lie $\infty$-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 examples are spelled out in detail: the $(-1)$-brane, ordinary gauge theory, higher gauge theory. Using these we demonstrate that the BV-formalism arises naturally from a construction of configuration space from an internal hom-object following in spirit, but not in detail, the very insightful [AKSZ, Roytenberg] (discussed previously).

You may think this blog is technical, but it’s not! We try to focus on broad-brush issues, not the niggling technical details that math research always seems to drag you down into.

But now I’m desperate: I want to finish a paper with Danny Stevenson on the classifying space of a topological 2-group, and the only remaining wrinkle seems to involve ‘strong NDR pairs’. Help!

We’re back! In the fall quarter of the Geometric Representation Theory Seminar, James Dolan and I developed the basic idea of groupoidification. In the winter quarter we’ll apply it to examples, starting with three closely related ones:

• the $q$-deformed harmonic oscillator,
• the Hall algebra of a quiver,
• the Hecke algebra of a Dynkin diagram.

Last time in part IX (part I, II, III, IV, V, VI, VII, VIII) I finally started moving from discussion of the purely differential graded $\infty$-algebraic structure underlying BV-quantization towards those ingredients which make BV theory into BV theory: the BV-Laplacian, the antibracket, the master equation.

In the last installment I had reviewed Witten’s old (but not at all particularly wide spread, it seems) nice observation which indicated that all this new structure is just old familiar structure in unusual guise.

This time I want to add yet another facet to that. I had complained before at various places in our BV-discussions here that I am not entirely fond of the currently very popular perspective on BV-formalism in terms of supermanifolds. I said: if we are really talking about Lie $\infty$-algebroids, then it seems awkward to model all our internal imagery on supergeometry, just because the Chevalley-Eilenberg algebra of any Lie $\infty$-algebra happens to be that: a graded algebra. Instead, we should use Lie-algebraic imagery.

To add substance to this vague idea, I’ll here go through the standard constructions of the antifields- and antighosts- and anti-ghosts-of-ghosts-, etc.-parts, which is usually thought of as forming the cotangent bundle $T^* X$ of the supermanifold $X$ of physical configurations, by using instead the Lie $\infty$-algebraic point of view which we invoke in Lie $\infty$-connections and applications to String- and Chern-Simons $n$-transport, combined with the Clifford-algebraic point of view that Witten highlighted.

In this spirit I will

- identify the configuration space $X$ as the action Lie $n$-algebroid $(g,V)$ (here $g$ denotes an $L_\infty$-algebra and $V$ a module for it) obtained from the $L_\infty$-algebra $g$ of physical symmetries, symmetries of symmetries, etc., acting on the space $V$ of fields, whose dual algebra is the Chevalley-Eilenberg algebra $CE(g,V)$ (definition 2);

- identify the shifted tangent bundle $T X$ with the inner automorphism Lie $(n+1)$-algebroid $inn(gg,V)$, corresponding to the tangent category of the groupoid integrating $(g,V)$, whose dual algebra is the Weil algebra $$W(g,V) = CE(inn(g,V))$$ (definition 5, section 4.1.1)

- identify the shifted cotangent bundle, dually with the Clifford algebra generated by $CE(gg,V)$, which is like differential forms on $X$ together with the horizontal inner derivations on $\mathrm{W}(g)$.

Here “horizontal” is with respect to the universal $(g,V)$-bundle which dually reads $$\array{ CE(g,V) \\ \uparrow \\ \mathrm{W}(g,V) \\ \uparrow \\ inv(g,V) }$$ (table 1)

- identify the inner pairing (often addressed as the graded symplectic pairing in the supermanifold imagery) of these (pairing of fields with anti-fields, ghosts with anti-ghosts, ghosts-of-ghosts with anti-ghosts-of-ghost, etc) with the co-adjoint action of horizontal vector fields on vertical vertor fields in the universal $(g,V)$-bundle, which means in symbols that $$(\iota_X, \omega) := L_{\iota_X} \omega = [[d_{\mathrm{W}(g,V)}, \iota_X], \omega] \,.$$

If you are an expert on BV-formalism in supermanifold language, you’ll find nothing new here after you unwrap my ideosyncratic terminology. Still I think this is worthwhile. The main change in perspective is:

instead of thinking of BV-formalism as living in the cotangent bundle of a supermanifold of physical configurations, we realize it as living in the horizontal derivations on the universal groupoid $n$-bundle of the action $n$-groupoid of gauge transformations acting on physical fields.

As 2-toposes seem to be cropping up a bit, here and here, let’s see if we can attract some experts to teach us about them.

On p. 36 of Mark Weber’s Strict 2-toposes, a 2-topos is defined as a finitely complete cartesian closed 2-category equipped with a duality involution and a classifying discrete opfibration. Cat is a good example of a 2-topos. Are there other familiar ones?

In

G. Ginot & M. Stiénon
Groupoid extensions, principal 2-group bundles and characteristic classes
arXiv:0801.1238

the authors regard principal 2-bundles on $X$ for any strict 2-group $G_{(2)} := (H \stackrel{t}{\to} G)$ in terms of their descent data/$G_{(2)}$-cocycles/transport anafunctors $$g : X \leftarrow [Y] \rightarrow \mathbf{B} G_{(2)} \,.$$

(Here $\mathbf{B}G_{(2)}$ is my notation for the one-object 2-groupoid defined by the 2-group $G_{(2)}$.)

Then they point out two things:

a) they demonstrate that and describe explicitly and in detail how such anafunctors, for $G_{(2)} = AUT(H) := (H \to \mathrm{Aut}(H))$ the automorphism 2-group of an ordinary group $H$, are equivalent to $H$-extensions of groupoids (such $H$-extensions of groupoids are a popular way to think of categorified bundles, as such usually addressed as “(bundle) gerbes” (What is the fiber??).)

b) they define a straightforward generalization of the notion of characteristic classes of principal 1-bundle to principal 2-bundle and prove that in the abelian case these characteristic classes of 2-bundles coincide with the familiar Dixmier-Douady classes known from bundle gerbes.

Guest post by Mike Stay

The Yoneda embedding is familiar in category theory. The continuation passing transform is familiar in computer programming.

They’re the same thing! Why doesn’t anyone ever say so?

In this, the final lecture of the fall’s Geometric Representation Theory seminar, I tried to wrap up by giving a correct statement of the Fundamental Theorem of Hecke Operators.

The fall seminar was a lot of fun, and very useful. It didn’t go the way I expected. I thought I thoroughly understood groupoidification, but I didn’t! So, all hell broke loose when I tried to state the Fundamental Theorem. The seminar threatened to swerve out of control, and Jim had to invent some more math to save the day. We skidded to safety at the very last second… but in the process, we learned a lot.

Will next quarter’s seminar be less hair-raising? Only time will tell!

I’ll talk about the topos-theoretic approach to the notion of the space of states of a physical system, recall the proposed answer by toposophers Döring, Isham, Landsman and others, suggest a simplified proposal and discuss it for the generalized $\sigma$-model class of physical systems which I am referring to as the charged quantum $n$-particle.

I’ll start with a detailed introduction that is supposed to make the discussion self-contained. The contribution that I would like to really discuss here with $n$-Café-readers is the last third, which starts with the paragraph title A topos theoretic state object for the charged $n$-particle?

This quarter I’m teaching a graduate math course on classical mechanics, focusing on Hamiltonian methods and symplectic geometry.

To get the course started, I’ll spend a class sketching the history of mechanics from Aristotle to Newton. It’s a hopeless task, but fun anyway… here are are my notes.

Guest post by Tom Leinster

Mark Weber has a new paper out, Familial 2-functors and parametric right adjoints. Among other things, it extends and improves some unpublished work of mine. Here I’ll explain just the part of Mark’s paper that I already knew about. It totally changed my attitude to simplicial sets and nerves.

For me, the moral of the story is this:

The nerve construction is inherent in the theory of categories.

A bit more precisely:

The category $\Delta$ and the nerve construction arise canonically from the free category monad on directed graphs.

By the ‘nerve construction’ I mean the usual functor $N: \mathbf{Cat} \to [\Delta^{op}, \mathbf{Set}]$, from small categories to simplicial sets.

I’ll start by reviewing the nerve construction. Then I’ll explain why for a long time I didn’t accept it as something natural — and why I finally did accept it. I’ll also give some examples and write a few words about Mark’s new work.

In the penultimate lecture of last fall’s Geometric Representation Theory seminar, James Dolan lays the last pieces of groundwork for the Fundamental Theorem of Hecke Operators.

I was looking at

J. F. Martins and T. Porter
On Yetter’s invariants and an extension of the Dijkgraaf-Witten invariant to categorical groups
(tac)

which João Martins pointed me to in a comment to the entry BF-Theory as a higher gauge theory. (John discussed this paper here a while back).

On the train back home, this inspired me to write the following notes, which happen to be mostly about ordinary Dijkgraaf-Witten theory, but try to put the general context into perspective.

[Update: Typed notes on this topic are now here: On $\Sigma$-models and nonabelian differential cohomology]

So far, in my discussion of BV-formalism (part I, II, III, IV, V, VI, VII, VIII) I had concentrated on the nature and meaning of the underlying complex, without saying a word yet about the antibracket and the BV-Laplacian and the master equation.

I hadn’t mentioned that yet because it wasn’t clear to me yet what the big story here actually is. But now I might be getting closer.

Recall from the discussion in Transgression of $n$-Transport and $n$-Connections that

Every differential non-negatively graded commutative algebra is, essentially, the algebra $\Omega^\bullet(X)$ of differential forms on some space.

Now generalize this fact from the cotangent bundle $T^* X$ to the Clifford bundle $T^*X \oplus T X$ as suggested in Categorified Clifford Algebra and weak Lie n-Algebras and recently discussed again in weak Lie $\infty$-algebras:

then we want to find

A kind of algebras such that each of them is, essentially, the Clifford algebra of $T^* X \oplus T X$ on some space $X$.

Apparently, this kind of algebra is: BV-algebra.

Definition A BV-algebra is a graded commutative algebra $A$ with an operator $\Delta : A \to A$ such that $\Delta^2 = 0$ and such that the “derived bracket” or “antibracket” $$[a,b] := \Delta(a b) - \Delta(a) b +(-1)^{|a|} a \Delta(b)$$ is a Gerstenhaber bracket on $A$.

The key to seeing this is related to Clifford algebra has been noticed two decades ago in

E. Witten
A note on the antibracket formalism
Modern Physics Letters A, 5 7, 487 - 494
(pdf)

The punchline is:

Here in the $n$-Café we happen to talk about the various notions of generalized smooth spaces every now and then (last time starting here).

I was dreaming of having, at one point, a survey of the various definitions and their relations in our non-existent wiki. Luckily, while I was just dreaming, Andrew Stacey did it.

Andrew is an expert on the index theorem for Dirac operators on loop spaces (see his list of research articles), and for that work he needs to deal with generalized smooth structures that render loop space a smooth space.

Last time I visited Nils Baas in Trondheim I had the pleasure of talking quite a bit with Andrew. Ever since then I had planned to post something about the intriguing things about loop space Dirac operators he taught me, but never found the time (but see this comment).

Now recently he sent me a link to his new article, which gives a detailed survey of the various definitions of generalized smooth spaces, and a careful and detailed comparison between them:

A. Stacey
Comparative Smootheology
(pdf, arXiv)

Abstract. We compare the different definitions of “the category of smooth objects”.

A friend of mine, Brendan Larvor, and I are wondering whether it would be a good idea to stage a conference which would bring together philosophers of mathematics from different camps.

Brendan is the author of Lakatos: An Introduction, and someone who believes as I do that one of our most important tasks is the Lakatosian one of attempting to understand the rationality of mathematics through the history of its practice.

By contrast, a much more orthodox philosophical approach to mathematics in the English-speaking world, well represented in the UK, is to address the question of whether mathematics is reducible to logic. To gain an idea of the current state of play here, you can take a look at What is Neologicism? by Linsky and Zalta. You can see from the final sentence of section 1 that organisational issues, such as whether category theory is a good language for mathematics, are irrelevant to them.

Happy New Year’s Day! The winter session of our seminar will start on Tuesday January 8th. To get you warmed up in the meantime, let’s see the last three lectures of the fall’s session, leading up to the long-awaited Fundamental Theorem of Hecke Operators.

In lecture 18 of the Geometric Representation Theory seminar, I began explaining degroupoidification — the process of turning groupoids into vector spaces and spans of groupoids into linear operators. I started with the prerequisites: the zeroth homology of groupoids, and groupoid cardinality.