1 The multiplicative monoid \(\mathbb {N}^{\times }\)
The Arithmetic Site, introduced by Connes and Consani, is a geometric object — a topos equipped with a structure sheaf — whose points in the topos-theoretic sense are in natural bijection with the adèle class space of \(\mathbb {Q}\). Its construction begins with a single, elementary monoid: the positive integers under multiplication. Everything else — the tropical semiring, the presheaf topos, the structure sheaf, and ultimately the connection to the Riemann zeta function — is built on top of this one piece of data.
Let \(\mathbb {N}^{\times }\) denote the set \(\{ 1, 2, 3, \ldots \} \) of strictly positive integers equipped with ordinary multiplication. This is a commutative monoid with unit \(1\). In Lean we represent this as PNat with its multiplicative structure, via the abbreviation NPos.
The unique factorisation theorem says that \(\mathbb {N}^{\times }\) is the free commutative monoid on the set of primes: every element factors uniquely as a finite product \(\prod _p p^{a_p}\). This arithmetic richness — the interplay of primes, divisibility, and factorisation — is what makes \(\mathbb {N}^{\times }\) the right monoid for the Arithmetic Site. As we will see, it is ultimately responsible for the fact that the points of \(\widehat{\mathbb {N}^{\times }}\) are parametrised by the adèle class space of \(\mathbb {Q}\), and that the Hasse–Weil zeta function of the site recovers the Riemann zeta function.
The classifying category \(B\mathbb {N}^{\times }\) of \(\mathbb {N}^{\times }\) is the one-object category whose single object is \(*\) and whose morphisms \(* \to *\) are the elements of \(\mathbb {N}^{\times }\), with composition given by multiplication.
The passage from a monoid \(M\) to the one-object category \(BM\) is the standard delooping construction. A presheaf on \(BM\) — that is, a functor \(BM^{\mathrm{op}} \to \mathbf{Set}\) — is exactly a set \(X\) equipped with a left action of \(M\): each element \(n \in M\) acts as a function \(X \to X\), and the action is compatible with composition. (Left and right actions coincide here since \(\mathbb {N}^{\times }\) is commutative.) The presheaf topos \(\widehat{\mathbb {N}^{\times }}\) introduced in Chapter 3 is precisely the category of such sets-with-\(\mathbb {N}^{\times }\)-action, and it is in this topos that the Arithmetic Site lives.