Combinatorial homotopy theory by Richard Williamson

By Richard Williamson

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For any functor int ≤1 A, there is a unique strict monoidal functor can A such that the following diagram in the category of categories commutes. 7. Proof The unique possible recipe for a strictly monoidal functor can A fitting into the above commutative diagram is the following. (i) We define can(I 0 ) to be int(0), and for n ≥ 1 we define can(I n ) to be int(1) ⊗ · · · ⊗ int(1) . n n ) to be the arrow (ii) For 1 ≤ i ≤ n, we define can(fi,0 can(I n−1 ) can(I i−1 ) ⊗ i0 ⊗ can(I n−i ) can(I n ) of A.

The colours indicate 1-cubes which must be the same. 48 Let us think of the free-standing 3-cube in the following way. (0,1,1) (1,1,1) (0,1,0) (1,1,0) (0,0,1) (1,0,1) (0,0,0) (1,0,0) Then we will depict the horn 2,1 3 in the manner shown below — the bottom face of the cube is that which is missing when we assemble our 3-horn from this net. (0, 0, 0) (1, 0, 0) (0, 0, 0) (0, 1, 0) (1, 1, 0) (1, 0, 0) (0, 0, 1) (0, 1, 1) (1, 1, 1) (1, 0, 1) (0, 0, 1) (1, 0, 1) To give another example, we will depict the horn 33,1 as follows — the back face of the cube is that which is missing when we assemble our 3-horn from this net.

I) We denote I 0 , I 1 , and I 2 by 0, 1, and 2 respectively. (ii) We denote i0 and i1 by ι0 and ι1 respectively. (iii) We denote p by π. (iv) We denote Γ0 and Γ1 by γ0 and γ1 respectively. Let A be a category equipped with a monoidal structure (⊗, λ, ρ, α). Suppose that I 0 , I 1 , i0 , i1 , p, Γ0 , Γ1 defines an interval in A equipped with a contraction structure p, a lower connection structure Γ0 , and an upper connection structure Γ1 . Suppose moreover that Γ0 is compatible with p, and that Γ1 is compatible with p.

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