By Satya Deo

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Secondly, the key property of a functor, namely that the map induced by a composition is the composition of the 42 3. DIMENSION GROUPS induced maps, is not clear with this approach. 2. We begin by showing that the invariant Ds (Σ, σ) is covariant for s-bijective maps. Having established this, we next assume that our shifts of ﬁnite type are presented by graphs H and G and that the s-bijective map arises from a graph homomorphism, π. We have seen in the last section that Ds (ΣH , σ) and Ds (ΣG , σ) may be computed in terms of H and G.

16. Let G and H be graphs. A graph homomorphism θ : H → G is left-covering if it is surjective and, for every v in H 0 , the map θ : t−1 {v} → t−1 {θ(v)} is a bijection. Similarly, π is right-covering if it is surjective and, for every v in H 0 , the map θ : i−1 {v} → i−1 {θ(v)} is a bijection. The following result is obvious and we omit the proof. 17. If G and H are graphs and θ : H → G is a left-covering (or right-covering) graph homomorphism, then the associated map θ : (ΣH , σ) → (ΣG , σ) is an s-bijective (or u-bijective, respectively) factor map.

Notice that, for n ≥ 1, the Smale bracket of n with +∞ is +∞ and so {n} ∼ {+∞}. It follows that in the group Ds (Σ, σ), < [−∞, n] >=< [−∞, m] >, for every m, n ∈ Z and < {n} >=< {+∞} >, for every n in Z. Moreover, Ds (Σ, σ) is isomorphic to Z2 , with these elements as generators. The order is lexicographic. 4. The dimension group as a covariant functor The construction of the dimension group has various subtle, but interesting functorial properties. 2 of [10], in the implication 1 implies 2 only in the case m = 1.