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 welldefined 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 0simplices — 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 ‘pullpush’ 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.

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_11_27_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung

Streaming
video in QuickTime format; the URL is
Re: Geometric Representation Theory (Lecture 17)
That’s cool!
I would like to know if there is a more systematic way to understand the pullback: it should really be the adjoint functor to some pushforward functor.
By the way, the links to the lecture notes currently don’t work!