The n-Category Café

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