Building a Coalition

David Ward

Frank Adams 1,

As events of this week are showing, it is often necessary to take a collection of smaller objects (such as parliamentary political parties), and combine them in some way to form a larger, more useful (?) object (such as a government). From a political viewpoint this is a fairly recent phenomena. However, we shall see that such ideas have been used in group theory for a number of decades.

We shall consider presheaves of abelian groups defined on simplicial complexes of subgroups of a given group, and see how they can be used to form modules for a given group. Continuing our analogue, we shall see that the simplices of the simplicial complex are like constituencies, whilst the associated abelian groups are the elected MPs. The process of forming a global module for the group then takes a much more rigorous form than that of forming a government. 

No prior knowledge of simplicial complexes, module theory or politics will be required

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