By Richard Williamson

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3. 1, given a cubical set X, the set Xn is the image of the object I n of under X. The face map fni, Xn Xn−1 for some 1 ≤ i ≤ n and 0 ≤ ≤ 1 is the image of the arrow I n−1 of fi,n In under X. The degeneracy map dn Xn Xn−1 for some 1 ≤ i ≤ n is the image of the arrow dni In I n−1 of under X. 6 — then the face map fni,0 is the image under X of the arrow I i−1 ⊗ i0 ⊗ I n−i I n−1 of , and the face map fni,1 is the image under X of the arrow I i−1 ⊗ i1 ⊗ I n−i I n−1 of In . The degeneracy map din is the image under X of the arrow In of In I i−1 ⊗ p ⊗ I n−i I n−1 .

Can we construct a homotopy theory of cubical sets? If so, how closely does it resemble the classical homotopy theory of topological spaces? This will be the topic of the remainder of the course. Notes ˇ 1 For example, the first Cech cohomology group of the Warsaw circle is Z, the integers. 2 There is a notion of homotopy equivalence with respect to which the Warsaw circle is homotopy equivalent to the circle, as we might intuitively expect: this is an aspect of shape theory. Very closely related ideas are involved in the theory of ´etale homotopy types of topoi, the original examples being the petit ´etale topoi arising in algebraic geometry.

2. In proving these deeper results, we typically construct certain double homotopies, namely maps X × I2 Y for some spaces X and Y . Rather beautifully, it turns out that we can build all the double homotopies we need from only two maps. 3. 4 prevented 1 from being a Kan complex. Suppose that we have a path I1 f X in a space X. Pre-composing with the map I2 I ×p I gives a map I2 X with the following boundary, in which the vertical sides are constant paths. f f 54 This corresponds to the degenerate 2-cube d22 (x) for a 1-cube x of a cubical set X.

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