Let CW be the category of CW-complexes and cellular maps, let CW^{0} be the full subcategory of path connected CW-complexes and let CW^{1} 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} : CW^{0} → HoCW together with a natural transformation

emb_{1} : t_{<1} → t_{<∞},

where t_{<∞} : CW^{0} → 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∗} : H_{0}(t_{<1}K) → H_{0}(t_{<∞}K) is an isomorphism and H_{r}(t_{<1}K) = 0 for r ≥ 1. The details are as follows: For a path connected CW-complex K, set t_{<1}(K) = k^{0}, where k^{0} is a 0-cell of K. Let emb_{1}(K) : t_{<1}(K) = k^{0} → t_{<∞}(K) = K be the inclusion of k^{0} in K. Then emb_{1∗} is an isomorphism on H^{0} as K is path connected. Clearly H_{r}(t_{<1}K) = 0 for r ≥ 1. Let f : K → L be a cellular map between objects of CW^{0}. The morphism t_{<1}(f) : t_{<1}(K) = k^{0} → l^{0} = 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 emb_{1} 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 k^{0} and l^{0} do not know anything about f, so l^{0} need not be the image of k^{0} under f.) Since L is path connected, there is a path ω : I → L from l^{0} = ω(0) to f (k^{0}) = ω(1). Then H : {k^{0}} × I → L, H(k^{0}, t) = ω(t), defines a homotopy from

k^{0} → l^{0} → L to k^{0} → K →^{f} L.

Dimension n = 2: We will define a covariant truncation functor t_{<n} = t_{<2} : CW^{1} → HoCW

together with a natural transformation

emb_{2} : t_{<2} → t_{<∞},

where t_{<∞} : CW^{1} → HoCW is as above (only restricted to simply connected spaces), such that for all objects K, emb_{2∗} : H_{r}(t_{<2}K) → H_{r}(t_{<∞}K) is an isomorphism for r = 0, 1, and H_{r}(t_{<2}K) = 0 for r ≥ 2. For a simply connected CW-complex K, set t_{<2}(K) = k^{0}, where k^{0} is a 0-cell of K. Let emb_{2}(K) : t_{<2}(K) = k^{0} → t_{<∞}(K) = K be the inclusion as in the case n = 1. It follows that emb_{2∗} is an isomorphism both on H_{0} as K is path connected and on H_{1} as H_{1}(k^{0}) = 0 = H_{1}(K), while trivially H_{r}(t_{<2}K) = 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 emb_{2} is a natural transformation.