Geometric Representation Theory (Lecture 18)
Posted by John Baez
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 longawaited 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.

Lecture 18 (Nov. 29)  John Baez on groupoidification. Turning a group $G$ acting on a set $S$ into a groupoid, the weak quotient $S//G$. Turning a map between groups acting on sets into a functor between groupoids. Degroupoidification as a 2functor from the bicategory
$[finite groupoids, spans of finite groupoids, equivalences between spans]$
to the bicategory
$[finitedimensional vector spaces, linear operators, equations between linear operators]$
Turning a groupoid $X$ into a vector space, namely the zeroth homology of $X$ with coefficients in the field $k$, denoted $H_0(X,k)$. This is the free vector space on the set of isomorphism classes of objects of $X$. Cohomology as dual to homology. Example: the homology of the groupoid of finite sets is the polynomial ring $k[z]$, while its cohomology is the ring of formal power series, $k[[z]]$.
Turning a span of finite groupoids into a linear operator using the concept of ‘groupoid cardinality’. Heuristic introduction to groupoid cardinality. The cardinality of a groupoid $X$ is the sum over objects $x$, one from each isomorphism class, of the fractions $1/Aut(x)$, where $Aut(x)$ is the automorphism group of $x$.
A puzzle: what’s the cardinality of the groupoid of finite sets?

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_11_29_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Apoorva Khare

Streaming
video in QuickTime format; the URL is