The Euclidean geometry is a category with point positions as the objects and Euclidean transformations as the morphisms. In kinematics there are two Euclidean geometries: that of length and that of duration. They are in turn categories:

Length space is a category with point locations as the objects and Euclidean transformations of points as the morphisms. Duration space is a category with instants (chronations) as the objects and Euclidean transformations of instants as the morphisms.

A functor can be defined from length space to duration space: every point location mapped to an instant and every transformation of point locations mapped to a transformation of instants. The opposite functor can also be mapped. Since the functor and its opposite functors are inverses, the categories of length space and duration space are isomorphic.

Reference: *Categories for the Working Mathematician* by Saunders MacLaine, 2nd edition (1971) p.14

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