Ivancevic_Applied-Diff-Geom

120 Applied Differential Geometry: A Modern Introductiongories K, L, ... as objects, functors F, G : K ⇒ L as arrows, and naturaltransformations, like τ : F → ·G as 2–arrows.In a similar way, the arrows in a 3–category K 3 are 2–functors F 2 , G 2 , ...sending objects in K 2 to objects in L 2 , arrows to arrows, and 2–arrows to2–arrows, strictly preserving all the structure of K 2Af∨ α❘ B✒gF 2 ✲ F2 (A)F 2 (f)❘F 2 (α) F 2 (B).∨ ✒F 2 (g)The 2–arrows in K 3 are 2–natural transformations, like τ 2 : F 22·⇒ G 2 between2–functors F 2 , G 2 : K 2 −→ L 2 that sends each object in K 2 to anarrow in L 2 and each arrow in K 2 to a 2–arrow in L 2 , and satisfies naturaltransformation–like conditions. We can visualize τ 2 as a prism goingfrom one functorial picture of K 2 in L 2 to another, built using commutativesquares:AfF 2 ✒F 2 (A)F 2 (f)❘F 2 (α) F 2 (B)∨ ✒F 2 (g)∨ α❘ B ⇓τ 2 (A)τ 2 (B)✒g❅G G 2 (f)2 ❅❅❘ ❄G 2 (A) G 2 (α) ❘ ❄K 2G 2 (B)∨ ✒G 2 (g) L 2Similarly, the arrows in a 4–category K 4 are 3–functors F 3 , G 3 , ... sendingobjects in K 3 to objects in L 3 , arrows to arrows, and 2–arrows to 2–arrows,strictly preserving all the structure of K 3fF 3 (f)Aαψ> gβ❘B✒F 3 ✲F 3 (ψ) ❘F 3 (A) F 3 (α) > F 3 (β) F 3 (B)✒ F 3 (g)