By Richard Williamson

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**Extra resources for Combinatorial homotopy theory**

**Example text**

Iv) There is a single non-degenerate 2-cube of id I2 of 2, corresponding to the identity arrow I2 . (v) All n-cubes for n > 2 are degenerate. 8. We think of the cubical sets n for n ≥ 0 as combinatorial analogues of the topological n-cubes. Ignoring degenerate i-cubes, we may depict n as follows in low dimensions, thinking of the square in the picture of 2 as solid, and similarly thinking of the cube and the squares in the picture of 3 as solid. 9. By a morphism from a cubical set X0 to a cubical set X1 we will mean a natural transformation from X0 to X1 .

We can capture the fact that Γ0 has the boundary we are looking for by the observation that the following diagrams in spaces commute. 3. 2 — of an interval I = 1, I 1 , i0 , i1 in a category A equipped with a monoidal structure (⊗, 1, λ, ρ, α). Indeed, we define an upper connection structure upon I to be an arrow Γ0 I1 ⊗ I1 I1 such that the following diagrams in A commute. 4. Similarly, we can capture the fact that Γ1 has the boundary we are looking for by the observation that the following diagrams in spaces commute.

1. We may think about the category of cubes with connections in a slightly different way — the reader not familiar with simplicial sets may wish to skip onto the next section. Recall that a simplicial set is a presheaf on a category ∆ which can be defined as the full subcategory of the category of partially ordered sets whose objects are the partially ordered sets 0 n−1 1 n for n ≥ 0. 2. We can define in a similar way. For any n ≥ 0, let I n denote the set n {0, 1} equipped with the partial ordering defined (δ0 , .