Ivancevic_Applied-Diff-Geom

122 Applied Differential Geometry: A Modern Introductionemerges:0 − cell : x • x ∈ M;1 − cell : x •f ✲ • y f : x ≃ y ∈ M,f : [0, 1] → M, f : x ↦→ y, y = f(x), f(0) = x, f(1) = y;e.g., linear path: f(t) = (1 − t)x + ty;2 − cell : x •f∨ h❘ • y h : f ≃ g ∈ M,✒gh : [0, 1] × [0, 1] → M, h : f ↦→ g, g = h(f(x)),h(x, 0) = f(x), h(x, 1) = g(x), h(0, t) = x, h(1, t) = ye.g., linear homotopy: h(x, t) = (1 − t)f(x) + tg(x);3 − cell : x •hfj> gi❘• y j : h ≃ i ∈ M,✒j : [0, 1] × [0, 1] × [0, 1] → M, j : h ↦→ i, i = j(h(f(x)))j(x, t, 0) = h(f(x)), j(x, t, 1) = i(f(x)),j(x, 0, s) = f(x), j(x, 1, s) = g(x),j(0, t, s) = x, j(1, t, s) = ye.g., linear composite homotopy: j(x, t, s) = (1 − t) h(f(x)) + t i(f(x)).If M is a smooth manifold, then all included paths and homotopies needto be smooth. Recall that a groupoid is a category in which every morphismis invertible; its special case with only one object is a group.Category T TTopological n−category T T has:• 0–cells: topological spaces X• 1–cells: continuous maps Xf ✲ Y