The n-Category Café

$$[finite groupoids, spans of finite groupoids, equivalences between spans]$$

to the bicategory

$$[finite-dimensional vector spaces, linear operators, equations between linear operators]$$

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