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 as the morphisms. Duration space is a category with chronations as the objects and Euclidean transformations as the morphisms.

A functor can be defined from length space to duration space: every point location mapped to a point chronation and every transformation of point locations mapped to a transformation of point chronations. 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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