Topology

SageMath is used to produce these images.

Animation of path homotopies in Theorem 51.2 (Munkres)

Associativity 

identity ≃ map p sending 0 to 0,

1/2 to 1/4, 3/4 to 1/2, 1 to 1

Left identity

e0 ∗ i ≃ i

Inverse

i ∗ i¯ ≃ e0

Inverse of path

f ∗ f¯ ≃ ex0