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 HoCWn 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, emb1∗ : H0(t<1K) → H0(t<∞K) is an isomorphism and Hr(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 emb1∗ is an isomorphism on H0 as K is path connected. Clearly Hr(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(idK) = [idk0] 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

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that is

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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, emb2∗ : Hr(t<2K) → Hr(t<∞K) is an isomorphism for r = 0, 1, and Hr(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 emb2∗ is an isomorphism both on H0 as K is path connected and on H1 as H1(k0) = 0 = H1(K), while trivially Hr(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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