By Emily Riehl

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This group is torsion, so any map, such as αZ/2nZ , which factors through the quotient by its torsion subgroup is zero. But n 0 ∈ Z/2nZ, a contradiction. 6. The Riesz representation theorem can be expressed as a natural isomorphism of functors from the category cHaus of compact Hausdorff spaces and continuous maps to the category Ban of real Banach spaces and continuous linear maps. Let Σ : cHaus → Ban be the functor that carries a compact Hausdorff space X to the Banach space Σ(X) of signed Baire measures on X and sends a continuous map f : X → Y to the map µ → µ ◦ f −1 : Σ(X) → Σ(Y).

Proof. vi. 6. 3 define an equivalence of categories between the category of pointed sets and the category of sets and partial functions. The composite U(−)+ is the identity on Set∂ , so one of the required natural isomorphisms is the identity. There is a natural isomorphism η : 1Set∗ (U−)+ whose components η(X,x) : (X, x) → (X\{x} ∪ {X\{x}}, X\{x}) are defined to be the based functions that act as the identity on X\{x}. Consider the categories Matk and Vectfd k of k-matrices and finite-dimensional nonzero k-vector spaces together with an intermediate category Vectbasis whose objects are k finite-dimensional vector spaces with chosen basis and whose morphisms are arbitrary (not necessarily basis-preserving) linear maps.

As r increases. There is a category PCluster whose objects are persistent clusters and whose morphisms are functions of underlying sets f : X → Y that define morphisms in Cluster for each r ∈ [0, ∞). Carlsson and Mémoli prove that there is a unique functor FinMet → PCluster, which takes the metric space with two points of distance r to the persistent cluster with one cluster for t ≥ r and two clusters for 0 ≤ t < r and satisfies two other reasonable conditions; see [CM13] for the details. 23The same argument, with the nth homotopy group functor πn : Top∗ → Group in place of π1 , proves that any continuous endomorphism of an n-dimensional disk has a fixed point.