By I.G. Macdonald

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18. A metric space is a set X together with a function d : X × X → [0, +∞) ∪ {+∞}, satisfying the following conditions. ) d(x, z) ≤ d(x, y) + d(y, z) Condition (c) is interpreted in the obvious way when some value is infinite. If {Xα }α∈A is an indexed family of metric spaces we let α∈A Xα denote the metric space whose underlying set is α∈A Xα and where the metric is defined by d(x1 , x2 ) = dα (x1 , x2 ) if x1 , x2 ∈ Xα and dα denotes the metric on Xα . d(x1 , x2 ) = +∞ if x1 ∈ Xα , x2 ∈ Xβ , α = β.

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. Consequently, it will suffice to prove that Spt (ρ) is an equivalence of spectra.

We define a symmetric monoidal functor ρ : P˜ → T . For a given object n[P1 , . . , Pn ] with Pi an object of either A or B, let a = a(P1 , . . , Pn ) be the sum in A of the objects belonging to A, and let b = b(P1 , . . , Pn ) be the sum of objects belonging to B. The ordering on the sum is the one inherited from the ordering of the Pi ’s. ρ is now defined on objects by ρ(n[P1 , . . , Pn ]) = (a(P1 , . . , Pn ), b(P1 , . . , Pn )). Suppose we have a morphism in P˜ , from n[P1 , . . , Pn ] to m[P1 , .