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January 1, 2008

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 long-awaited 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 GG acting on a set SS into a groupoid, the weak quotient S//GS//G. Turning a map between groups acting on sets into a functor between groupoids. Degroupoidification as a 2-functor from the bicategory

    [finitegroupoids,spansoffinitegroupoids,equivalencesbetweenspans][finite groupoids, spans of finite groupoids, equivalences between spans]

    to the bicategory

    [finitedimensionalvectorspaces,linearoperators,equationsbetweenlinearoperators][finite-dimensional vector spaces, linear operators, equations between linear operators]

    Turning a groupoid XX into a vector space, namely the zeroth homology of XX with coefficients in the field kk, denoted H 0(X,k)H_0(X,k). This is the free vector space on the set of isomorphism classes of objects of XX. Cohomology as dual to homology. Example: the homology of the groupoid of finite sets is the polynomial ring k[z]k[z], while its cohomology is the ring of formal power series, k[[z]]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 XX is the sum over objects xx, one from each isomorphism class, of the fractions 1/|Aut(x)|1/|Aut(x)|, where Aut(x)Aut(x) is the automorphism group of xx.

    A puzzle: what’s the cardinality of the groupoid of finite sets?

Posted at January 1, 2008 6:45 PM UTC

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