Functoriality in Low Dimensions. Note Quote.

Let CW be the category of CW-complexes and cellular maps, let CW⁰ be the full subcategory of path connected CW-complexes and let CW¹ be the full subcategory of simply connected CW-complexes. Let HoCW denote the category of CW-complexes and homotopy classes of cellular maps. Let HoCW_n denote the category of CW-complexes and rel n-skeleton homotopy classes of cellular maps. Dimension n = 1: It is straightforward to define a covariant truncation functor

t_<n = t_<1 : CW⁰ → HoCW together with a natural transformation

emb₁ : t_<1 → t_<∞,

where t_<∞ : CW⁰ → HoCW is the natural “inclusion-followed-by-quotient” functor given by t_<∞(K) = K for objects K and t_<∞(f) = [f] for morphisms f, such that for all objects K, emb_1∗ : H₀(t_<1K) → H₀(t_<∞K) is an isomorphism and H_r(t_<1K) = 0 for r ≥ 1. The details are as follows: For a path connected CW-complex K, set t_<1(K) = k⁰, where k⁰ is a 0-cell of K. Let emb₁(K) : t_<1(K) = k⁰ → t_<∞(K) = K be the inclusion of k⁰ in K. Then emb_1∗ is an isomorphism on H⁰ as K is path connected. Clearly H_r(t_<1K) = 0 for r ≥ 1. Let f : K → L be a cellular map between objects of CW⁰. The morphism t_<1(f) : t_<1(K) = k⁰ → l⁰ = t_<1(L) is the homotopy class of the unique map from a point to a point. In particular, t_<1(id_K) = [id_k⁰] and for a cellular map g : L → P we have t_<1(gf) = t_<1(g) ◦ t_<1(f), so that t_<1 is indeed a functor. To show that emb₁ is a natural transformation, we need to see that

that is

commutes in HoCW. This is where we need the functor t_<1 to have values only in HoCW, not in CW, because the square need certainly not commute in CW. (The points k⁰ and l⁰ do not know anything about f, so l⁰ need not be the image of k⁰ under f.) Since L is path connected, there is a path ω : I → L from l⁰ = ω(0) to f (k⁰) = ω(1). Then H : {k⁰} × I → L, H(k⁰, t) = ω(t), defines a homotopy from

k⁰ → l⁰ → L to k⁰ → K →^f L.

Dimension n = 2: We will define a covariant truncation functor t_<n = t_<2 : CW¹ → HoCW

together with a natural transformation
emb₂ : t_<2 → t_<∞,

where t_<∞ : CW¹ → HoCW is as above (only restricted to simply connected spaces), such that for all objects K, emb_2∗ : H_r(t_<2K) → H_r(t_<∞K) is an isomorphism for r = 0, 1, and H_r(t_<2K) = 0 for r ≥ 2. For a simply connected CW-complex K, set t_<2(K) = k⁰, where k⁰ is a 0-cell of K. Let emb₂(K) : t_<2(K) = k⁰ → t_<∞(K) = K be the inclusion as in the case n = 1. It follows that emb_2∗ is an isomorphism both on H₀ as K is path connected and on H₁ as H₁(k⁰) = 0 = H₁(K), while trivially H_r(t_<2K) = 0 for r ≥ 2. On a cellular map f, t_<2(f) is defined as in the case n = 1. As in the case n = 1, this yields a functor and emb₂ is a natural transformation.

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