Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 97Many properties of mathematical constructions may be represented byuniversal properties of diagrams [MacLane (1971)]. Consider the Cartesianproduct X × Y of two sets, consisting as usual of all ordered pairs 〈x, y〉 ofelements x ∈ X and y ∈ Y . The projections 〈x, y〉 ↦→ x, 〈x, y〉 ↦→ y of theproduct on its ‘axes’ X and Y are functions p : X ×Y → X, q : X ×Y → Y .Any function h : W → X × Y from a third set W is uniquely determinedby its composites p ◦ h and q ◦ h. Conversely, given W and two functionsf and g as in the diagram below, there is a unique function h which makesthe following diagram commute:W ❅f h ❅ g❅❅❘✠ ❄X ✛ p X × Y ✲q YThis property describes the Cartesian product X × Y uniquely; the samediagram, read in the category of topological spaces or of groups, describesuniquely the Cartesian product of spaces or of the direct product of groups.The construction ‘Cartesian product’ is technically called a ‘functor’because it applies suitably both to the sets and to the functions betweenthem; two functions k : X → X ′ and l : Y → Y ′ have a function k × l astheir Cartesian product:k × l : X × Y → X ′ × Y ′ ,〈x, y〉 ↦→ 〈kx, ly〉.2.3.1.12 Groups and Related Algebraic StructuresAs already stated, the basic functional unit of lower biodynamics is thespecial Euclidean group SE(3) of rigid body motions. In general, a groupis a pointed set (G, e) with a multiplication µ : G × G → G and an inverseν : G → G such that the following diagrams commute [Switzer (1975)]:(1)G(e is a two–sided identity)(e, 1)✲G × G(1, e)✲ G❅1 ❅❅❅❘ µ 1❄ ✒G