By Mumford.

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Pn ] to m[P1 , . . , Pm ]. Let U denote the direct sum of all the Pi ’s for which i = f , and let V denote the direct sum of all the Pi ’s for which i = g. Then there is determined an isomorphism from a(P1 , . . , Pn ) to U ⊕ a(P1 , . . , Pm ) ⊕ V , and an isomorphism uU ⊕ b(P1 , . . , Pn ) ⊕ vV → b(P1 , . . , Pm ). This defines a morphism in T , which is defined to be the image under ρ of the given morphism in P˜ . Observe that ρ and θ are naturally isomorphic symmetric monoidal functors; 24 Gunnar Carlsson the isomorphism involves only repeated application of the isomorphic symmetric OA ⊕ a ∼ = a and OB ⊕ b ∼ = b coming from the unital structure on A and B, where OA and OB denote the zero objects of A and B, respectively.

Let X. denote the cosimplicial Γ–space defining holim F | C. )hΓ are fibrant cosimplicial spaces. )Γ → (X . )hΓ is a weak  Γ  hΓ equivalence, hence by (a), holim F | C  → holim F | C  ←− C ←− C is a weak equivalence. Proof: (a) is direct, entirely analogous to the proof that the cosimplicial space defining holim is fibrant if F (x) is a Kan complex for all x ∈ C. We ←− C leave it to the reader. To prove (b), we must show that if C is discrete category with free Γ–action, and Γ C → s–sets is a functor, then the Γ  hΓ  natural map holim F | C  → holim F | C  ←− C ←− C is a weak equivalence.

Moreover, since by hypothesis, any closed ball is compact, we see that if ξ is locally finite, the set {σ ∈ supp (ξ) | image (σ) ∩ BR (x)} is finite for all R and x. Consequently, if ξ is small of order Ur for some r, it lies in bˆ C∗ (X; G). Let Cˆ∗Ur (X; G) denote the subcomplex of Cˆ∗ (X; G) consisting of chains small of order Ur . We have shown that b Cˆ∗ (X; G) = Cˆ∗Ur (X; G) ⊆ Cˆ∗ (X; G) . r ˆ If we can show that each inclusion Cˆ∗Ur (X; G) → C(X; G) induces an isomorphism on homology, then the result will follow.

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