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The mapping class group of a surface with one boundary component admitsnumerous interesting representations including as a group of automorphisms of afree group and as a group of symplectic transformations. Insofar as the mappingclass group can be identified with the fundamental group of Riemann's modulispace, it is furthermore identified with a subgroup of the fundamental pathgroupoid upon choosing a basepoint. A combinatorial model for this, the mappingclass groupoid, arises from the invariant cell decomposition of Teichm\"ullerspace, whose fundamental path groupoid is called the Ptolemy groupoid. It isnatural to try to extend representations of the mapping class group to themapping class groupoid, i.e., construct a homomorphism from the mapping classgroupoid to the same target that extends the given representations arising fromvarious choices of basepoint.  Among others, we extend both aforementioned representations to the groupoidlevel in this sense, where the symplectic representation is lifted bothrationally and integrally. The techniques of proof include several algorithmsinvolving fatgraphs and chord diagrams. The former extension is given byexplicit formulae depending upon six essential cases, and the kernel and imageof the groupoid representation are computed. Furthermore, this providesgroupoid extensions of any representation of the mapping class group thatfactors through its action on the fundamental group of the surface including,for instance, the Magnus representation and representations on the modulispaces of flat connections.

Groupoid Extensions of Mapping Class Representations for Bordered Surfaces