Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 95γ : I → X with γ(0) = x 0 and γ(1) = x 1 . Thus X I is the space of all pathsin X with the compact–open topology. We introduce a relation ∼ on X bysaying x 0 ∼ x 1 iff there is a path γ : I → X from x 0 to x 1 . ∼ is clearly anequivalence relation, and the set of equivalence classes is denoted by π 0 (X).The elements of π 0 (X) are called the path components, or 0−componentsof X. If π 0 (X) contains just one element, then X is called path connected,or 0−connected. A closed path, or loop in X at the point x 0 is a path γ(t)for which γ(0) = γ(1) = x 0 . The inverse loop γ −1 (t) based at x 0 ∈ X isdefined by γ −1 (t) = γ(1 − t), for 0 ≤ t ≤ 1. The homotopy of loops is theparticular case of the above defined homotopy of continuous maps.If (X, x 0 ) is a pointed space, then we may regard π 0 (X) as a pointed setwith the 0−component of x 0 as a base point. We use the notation π 0 (X, x 0 )to denote p 0 (X, x 0 ) thought of as a pointed set. If f : X → Y is a map thenf sends 0−components of X into 0−components of Y and hence defines afunction π 0 (f) : π 0 (X) → π 0 (Y ). Similarly, a base point–preserving mapf : (X, x 0 ) → (Y, y 0 ) induces a map of pointed sets π 0 (f) : π 0 (X, x 0 ) →π 0 (Y, y 0 ). In this way defined π 0 represents a ‘functor’ from the ‘category’of topological (point) spaces to the underlying category of (point) sets (seethe next section).Combination of topology and calculus gives differential topology, or differentialgeometry.2.3.1.11 Commutative DiagramsThe category theory (see below) was born with an observation that manyproperties of mathematical systems can be unified and simplified by a presentationwith commutative diagrams of arrows [MacLane (1971)]. Eacharrow f : X → Y represents a function (i.e., a map, transformation, operator);that is, a source (domain) set X, a target (codomain) set Y , and arule x ↦→ f(x) which assigns to each element x ∈ X an element f(x) ∈ Y .A typical diagram of sets and functions isXf✲ YXf ✲ f(X)❅❅❅❅❘Zhg❄or❅h ❅❅❅❘g(f(X)) g❄This diagram is commutative iff h = g◦f, where g◦f is the usual compositefunction g ◦ f : X → Z, defined by x ↦→ g(f(x)).