Abstract

Generalizing the algebro-geometric two-dimensional complex plane and Hamilton’s quaternions to the setting of an Euclidean vector space E of arbitrary finite dimension leads to the notion of a Clifford algebra of E, its even subalgebras and its spin group with its natural action on E by orthogonal transformations. This construction is remarkable in two ways. On the one hand the Clifford algebra provides a synthesis of the Euclidean synthetic and Cartesian analytic approach to geometry and so becomes a geometric algebra in the spirit of Leibniz. On the other hand this functorial way of algebraizing geometry is manifestly covariant, as opposed to the standard contravariant way of mapping a space to its ring of ‘coordinate functions’.

One of the most important contributors to the field of Geometric Algebra was H.G. Grassmann with his theory of linear extension. Surprisingly, in his original work Grassmann defined his exterior algebra not only for vector spaces, but foremost for affine spaces. Being concerned with the foundations of Geometric Algebra this leads to the natural question: “What is the geometric algebra of an affine space?”

In my talk I will attempt to give several possible answers to this question using Category Theory as a particular form of conceptual mathematics as well as the two leading examples of a Grassmann and Clifford algebra of an affine space. Ranging from the general to the particular these are: the category of augmented algebras, like e.g. Hopf algebras, certain differential graded algebras, and, following a suggestion of Lawvere, the category of dynamical algebras.