By Satya Deo

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Ii) For 1 ≤ i ≤ m and n ≥ 0, and 0 ≤ ≤ 1, we define fi,m ⊗ I n to be fi,m+n . m+n (iii) For 1 ≤ i ≤ n and m ≥ 0, and 0 ≤ ≤ 1, we define I m ⊗ fi,n to be fm+i, . m+n n (iv) For 1 ≤ i ≤ m and n ≥ 0, we define dm . i ⊗ I to be di (v) For 1 ≤ i ≤ n and m ≥ 0, we define I m ⊗ dni to be dm+n m+i . m+n n (vi) For any m ≥ 2, 1 ≤ i ≤ m, n ≥ 0, and 0 ≤ ≤ 1, we define Γm . i, ⊗ I to be Γi, (vii) For any n ≥ 2, 1 ≤ i ≤ n, m ≥ 0, and 0 ≤ ≤ 1, we define I m ⊗ Γni, to be Γm+n m+i, . (viii) For m, m , m , n ≥ 0, and arrows g0 Im Im and g1 Im of Im , we define (g1 ◦ g0 ) ⊗ I n to be (g1 ⊗ I n ) ◦ (g0 ⊗ I n ).

Then we have that x0 ∼ x1 by virtue of the 2-cube x, but since there is no 2-cube with the following boundary x1 ∗ ∗ ∗ ∗ x0 we do not have that x1 ∼ x0 . 10. Morevoer ∼ is not transitive. For a minimal counter-example, let X denote the cubical set uniquely defined by the following recipe. (i) We have a single 0-cube ∗. (ii) We have exactly three non-degenerate 1-cubes x0 , x1 , and x2 , all with the following boundary. ∗ ∗ (iii) We have exactly two non-degenerate 2-cubes: one, which we will denote by x, with the boundary depicted below x0 ∗ ∗ x1 ∗ ∗ and one, which we will denote by x , with the boundary depicted below.

2. We will give the details later in the course. Let (X, ∗) be a pointed cubical set, and let x0 and x1 be n-cubes of X belonging to Zn (X, ∗) — intuitively, n-cubes whose boundary is trivial. For now, let us take it on faith that a homotopy from the morphism In x0 X of cubical sets to the morphism In x1 X of cubical sets is exactly a morphism I n+1 h X of cubical sets such that the following diagram in Set In I n ⊗ i0 x0 In I n+1 h op I n ⊗ i1 x1 X commute I n+1 h X 42 and moreover the following diagrams in Set In I i−1 ⊗ i0 ⊗ I n−i p op commute for every 1 ≤ i ≤ n.