By Richard Williamson

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**Extra resources for Combinatorial homotopy theory**

**Example text**

A profound insight of Kan — which goes back to the paper [4] — is that although ∼ may not define an equivalence relation for arbitrary cubical sets, it will define an equivalence relation for cubical sets satisfying a certain extension property, now known as Kan complexes. It is to introducing these gadgets that we now turn. 2. 1. 1. We have become acquainted with the free-standing n-cubes n . Let us now define a couple more families of cubical sets which will be of vital importance to us. 2. Definition For any n ≥ 1, we denote by ∂ n the cubical set obtained as follows, and refer to it as boundary of the free-standing n-cube.

8, it follows immediately from the fact that (⊗, 1) defines a strict monoidal structure upon A that the defining relations of give rise under can(−) to commutative diagrams in A. 8. 1. Definition A cubical set is a presheaf on . 2. 3. 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.

If an n-cube of X is not obtainable as din (x) for some (n − 1)-cube x of X and some 1 ≤ i ≤ n, we refer to it as non-degenerate. 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 .