4 The structure sheaf
In classical algebraic geometry, a ringed space is a topological space equipped with a sheaf of commutative rings — the structure sheaf — whose stalks encode the local algebra of the space. The Arithmetic Site is a ringed topos in the same spirit: the role of the topological space is played by the presheaf topos \(\widehat{\mathbb {N}^{\times }}\), and the role of the structure sheaf is played by \(\bar{\mathbb {N}}\), viewed as a semiring object living inside that topos.
An object of \(\widehat{\mathbb {N}^{\times }}\) is a set equipped with a left \(\mathbb {N}^{\times }\)-action, so for \(\bar{\mathbb {N}}\) to be a semiring object of \(\widehat{\mathbb {N}^{\times }}\) the action must be by semiring endomorphisms: each scaling map \(\varphi _n\) must preserve both tropical operations and both identities. This is exactly the content of Lemma 2.6, which is therefore the key structural fact underlying the definition below.
The structure semiring \(\mathcal{O}\) of the Arithmetic Site is the commutative-semiring-valued presheaf whose value is \(\bar{\mathbb {N}}\) and whose restriction maps are the scaling endomorphisms \(n \cdot x := \varphi _n(x)=nx\).
The underlying Type-valued structure sheaf is obtained from \(\mathcal O\) by forgetting its commutative-semiring structure pointwise.
The assignment \(n \mapsto \varphi _n\) defines a monoid homomorphism \((\mathbb {N}^{\times }, \times ) \to \mathrm{End}_{\mathrm{Semiring}}(\bar{\mathbb {N}})\), making \(\bar{\mathbb {N}}\) into a semiring object of \(\widehat{\mathbb {N}^{\times }}\) with \(\mathbb {N}^{\times }\) acting by multiplication.
Each \(\varphi _n\) is a semiring endomorphism by Lemma 2.6. The homomorphism property \(\varphi _{mn} = \varphi _m \circ \varphi _n\) is Lemma 2.7, and \(\varphi _1 = \mathrm{id}\) since \(\varphi _1(x) = 1 \cdot x = x\). Lean packages both laws in scalingMonoidHom, using scalingEndomorphism_comp for multiplication.
Since the Arithmetic Site uses the chaotic topology, every presheaf on \(B\mathbb {N}^{\times }\) is automatically a sheaf — there are no gluing conditions to verify. The content of the structure sheaf is therefore entirely algebraic: \(\bar{\mathbb {N}}\) is not merely a set with \(\mathbb {N}^{\times }\)-action, but a semiring whose operations are preserved by that action. This is what Lemma 4.3 records.
Connes and Consani (Theorem 2.4 of their 2014 paper) describe the stalks of \(\mathcal{O}\) explicitly. At the point corresponding to an intermediate semifield \(F \subset K \subset \bar{F}\), the stalk is
the subsemiring of elements that are “bounded above by \(1\)” in the ordering of \(K\). This is the characteristic-\(1\) analogue of the ring of integers of a local field: just as the ring of integers of a \(p\)-adic field \(K_v\) consists of elements of norm at most \(1\), the stalk of \(\mathcal{O}\) picks out the elements “bounded above by \(1\)”. At the “generic” point (corresponding to \(K = F = Z_{\max }\)) the stalk recovers \(\bar{\mathbb {N}}\) itself.