December 26, 2007

Geometric Representation Theory (Lecture 17)

Posted by John Baez This time in the Geometric Representation Theory seminar, James Dolan explains ‘degroupoidification’ — the process of turning a span of groupoids into a linear operator between vector spaces. We’ve been telling people about this for a while now, for example in week256 of This Week’s Finds. But now Jim reveals more about what’s really going on.

It’s all about something topologists call ‘transfer’. Taking the homology of a space is a covariant thing to do: given a map of spaces

$f: X \to Y$

we get a map of homology groups

$f_* : H_*(X) \to H_*(Y)$

But, in some special situations homology is also contravariant: we also get a map going backwards, called the ‘transfer’:

$f^! : H_*(Y) \to H_*(X)$

The exclamation mark, pronounced ‘shriek’ here, is a hint that something perverse and shocking is going on!

Transfer is only well-defined when the map $f$ is nice, for example a finite covering map. Recall that elements of $H_*(X)$ are linear combinations of equivalence classes of simplices in $X$. We define $f_*$ in the obvious way, by sending any simplex in $X$ to a simplex in $Y$, its image under $f$. But when $f$ is a finite covering map, each simplex in $Y$ is the image of finitely many simplices in $X$. So, we can define $f^!$ by sending any simplex in $Y$ to a sum of simplices in $X$: all its inverse images under $f$.

So far I’ve been talking about transfer for homology of spaces. But degroupoidification involves transfer for the zeroth homology of groupoids. If $X$ is a finite groupoid, $H_0(X)$ consists of formal linear combinations of isomorphism classes of objects in $X$. (A groupoid is like a space, and objects are like 0-simplices — that is, points.) If

$f: X \to Y$

is a functor, we get a linear operator

$f_* : H_0(X) \to H_0(Y)$

in an obvious way, by sending any object in $X$ to an object in $Y$, its image under $f$. But, if $X$ and $Y$ are finite groupoids, we also get a linear operator going the other way, called the ‘transfer’:

$f^!: H_0(Y) \to H_0(X)$

by sending any object in $Y$ to a cleverly weighted sum of objects in $X$: all its inverse images under $f$.

The ‘clever weighting’ will involve the concept of ‘groupoid cardinality’, to be introduced shortly.

With transfer in hand, we can turn a span of finite groupoids into a linear operator: the span

$X \stackrel{f}{\leftarrow} S \stackrel{g}{\rightarrow} Y$

turns into the linear operator built by composing $f^!: H_0(X) \to H_0(S)$ with $g_*: H_0(S) \to H_0(Y)$. This is sometimes called a ‘pull-push’ construction, since we pull back along $f$ and then push forwards along $g$.

• Lecture 17 (Nov. 27) - James Dolan on degroupoidification. The 0th homology of a groupoid. Why groupoids don’t get enough respect. Why 0th homology doesn’t get enough respect. Transfer maps for 0th homology.

Posted at December 26, 2007 4:32 PM UTC

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Re: Geometric Representation Theory (Lecture 17)

That’s cool!

I would like to know if there is a more systematic way to understand the pull-back: it should really be the adjoint functor to some push-forward functor.

By the way, the links to the lecture notes currently don’t work!

Posted by: Urs Schreiber on December 27, 2007 2:53 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

I too found the link didn’t work. But it does if you start QuickTime and then enter the URL.

Posted by: David Corfield on December 27, 2007 2:59 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

I too found the link didn’t work. But it does if you start QuickTime and then enter the URL.

Hm, I meant the PDFs containing the notes by Alex Hoffnung and Apoorva Khare.

Apart from technological constraints here at home, I feel very conservative and enjoy absorbing this stuff by staring at a document more than by watching a video.

Posted by: Urs Schreiber on December 27, 2007 3:22 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

OK, I fixed the links.

I imagine the information flow in a 75 minute session is rather slow for you. Perhaps a student could be set the task of editing 15 minutes of highlights each week.

Posted by: David Corfield on December 27, 2007 3:46 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

The links still aren’t right as John has Apoorva Khare’s notes for his 29 Nov. lecture up rather than for Jim’s 27 Nov.

Posted by: David Corfield on December 27, 2007 4:05 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

I believe the problem is that Apoorva missed this lecture by Jim, so his “17th lecture notes” were actually his notes for the next lecture by me.

So: no notes from Apoorva this time.

I can understand how experts like Urs would find it more efficient to read this stuff rather than watch videos. Alas, it’s very hard for anyone to take notes of Jim’s lectures, since he talks more than writes. And, with the departure of Derek Wise, and the start of videotaping, I’m less inclined to write lots on the board myself.

But, there will be stuff to read.

The main result in this fall’s seminar was the modestly named ‘Fundamental Theorem of Hecke Operators’. A precise statement and proof should appear in a paper by me and my grad student Christopher Walker… he wants to write up this paper and talk about it for his thesis qualifying exam.

I may write up a rough draft in a week or so, minus the proof, to go along with the video of my last lecture, where I tried to state this result in a less technical way.

(The full precise statement involves topoi.)

Posted by: John Baez on December 28, 2007 2:26 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

Urs wrote:

I would like to know if there is a more systematic way to understand the pull-back: it should really be the adjoint functor to some push-forward functor.

Maybe that’s easier to understand before we degroupoidify. A functor between groupoids

$f: X \to Y$

gives rise to a pullback functor

$f^*: hom(Y,FinSet) \to hom(X,FinSet)$

and this has a left adjoint

$f_*: hom(X,FinSet) \to hom(Y,FinSet)$

You can think of an object in $hom(X,Set)$ as a ‘$FinSet$-linear combination of objects in $X$’. Similarly, an element of the homology $H_0(X)$ is a $\mathbb{C}$-linear combination of isomorphism classes of objects in $X$. So, they’re closely related: the latter is a kind of decategorification of the former. Indeed, a $FinSet$-linear combination of objects in $X$ should give a $\mathbb{C}$-linear combination of isomorphism classes of objects, at least when $X$ is finite so the necessary sums converge.

So, the pushforward and pullback in homology

$f_* : H_0(X) \to H_0(Y)$

$f^! : H_0(Y) \to H_0(X)$

should be something like decategorified versions of the functors I’ve listed above. However, I’m afraid that I’ll screw up if I say anything more precise before doing some calculations!

Posted by: John Baez on January 1, 2008 6:44 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

Presumably the composition of $g^!$ with $f_*$ gives you the transpose of the composition of $f^!$ with $g_*$.

Posted by: David Corfield on December 27, 2007 3:07 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 17)

Beck–Chevalley on your mind?

Posted by: John Baez on January 1, 2008 6:23 PM | Permalink | Reply to this

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