## December 13, 2007

### Geometric Representation Theory (Lecture 15)

#### Posted by John Baez

This time in the Geometric Representation Theory seminar, James Dolan begins the process of cleaning up the mess I made last time, when I was trying to state the ‘Fundamental Theorem of Hecke Operators’.

A quick and dirty fix is not hard. But a really beautiful and conceptually clear statement of this theorem takes more work. In fact, it’ll take the rest of this fall’s course! Jim begins by explaining why my statement was false. Then he turns to various forms of decategorification — as a warmup for the one we’ll need here, namely ‘degroupoidification’.

• Lecture 15 (Nov. 15) - James Dolan on the fundamental theorem of Hecke operators and various forms of decategorification. The problem with the statement from last time. Decategorification processes. Turning a category into a set: its set of isomorphism classes of objects. Turning a finite set into a natural number: its cardinality. Turning a finite-dimensional vector space into a natural number: its dimension. Another way to turn a category into a set: its set of components. $\pi_0$ turns a topological space into a set: its set of components. $\pi_{-1}$ turns a space into a truth value: the empty space become ‘false’, while nonempty spaces become ‘true’. The Grothendieck group construction turns an abelian category into an abelian group. Degroupoidification turns finite groupoids into finite-dimensional vector spaces, and spans into linear operators.

Posted at December 13, 2007 1:33 AM UTC

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