We will construct the circle purely as a sheaf on the Cartesian site (objects \(\mathbb{R}^n\), morphisms smooth maps, standard smooth coverings). We will construct a sheaf \(S^1\) from the representable sheaf of the real line \(h_{\mathbb{R}}\) and the integers \(\underline{\mathbb{Z}}\), check the descent properties, and explain the usual local picture found in the standard construction.
Let \(\mathbf{Cart}\) be the Cartesian site: objects \(\mathbb{R}^n\) (\(n\ge0\)), morphisms smooth maps, and the Grothendieck topology is given by the usual open covers (unions of open immersions). For any object \(X\in\mathbf{Cart}\) denote by
\[
h_X:=\mathrm{Hom}(-,X)
\]
a representable presheaf. In particular \(h_{\mathbb{R}}\) is a presheaf of smooth maps into the real line:
\[
h_{\mathbb{R}}(U)=\mathrm{C}^\infty(U,\mathbb{R}),\qquad U\in\mathbf{Cart}.
\]
We will work in the category \(\mathbf{Sh}(\mathbf{Cart})\) of sheaves of sets on this site.
\(\underline{\mathbb{Z}}\) and its action on \(h_{\mathbb R}\)
Let \(\underline{\mathbb{Z}}\) denote the constant sheaf associated to the discrete set \(\mathbb{Z}\). Concretely,
\[
\underline{\mathbb{Z}}(U)=\{ \text{locally constant functions } U\to\mathbb{Z} \}.
\]
(Equivalently the sheafification of the presheaf \(U\mapsto \mathbb{Z}\))
There is a natural action of \(\underline{\mathbb{Z}}\) on the representable sheaf \(h_{\mathbb R}\) given on sections by pointwise integer translation:
\[
\alpha_U:\underline{\mathbb{Z}}(U)\times h_{\mathbb{R}}(U)
\to h_{\mathbb{R}}(U),\qquad (n,f)\mapsto f+n,
\]
where \((f+n)(u):=f(u)+n(u)\) and \(n(u)\in\mathbb{Z}\) is locally constant. This defines a morphism of sheaves
\[
\alpha:\underline{\mathbb{Z}}\times h_{\mathbb{R}}\longrightarrow h_{\mathbb{R}}.
\]
\(h_{\mathbb R}/\underline{\mathbb Z}\)
Form the two parallel arrows of sheaves
\[
\underline{\mathbb{Z}}\times h_{\mathbb{R}}\rightrightarrows h_{\mathbb{R}}
\]
given by the projection \(\mathrm{pr}_2:(n,f)\mapsto f\) and the action \(\alpha:(n,f)\mapsto f+n\). Define \(S^1\) to be the coequalizer in \(\mathbf{Sh}(\mathbf{Cart})\):
\[
S^1 := \operatorname{coeq}\Big(\underline{\mathbb{Z}}\times h_{\mathbb{R}}\rightrightarrows h_{\mathbb{R}}\Big).
\]
Because \(\mathbf{Sh}(\mathbf{Cart})\) is a topos, coequalizers exist, so this object is well-defined. Concretely, the presheaf quotient \(U\mapsto h_{\mathbb R}(U)/\sim\) (where \(f\sim g\) iff there exists a locally constant \(n:U\to\mathbb{Z}\) with \(g=f+n\)) is already a sheaf (the equivalence relation is local), so the coequalizer as a presheaf is already a sheaf.
Thus for each \(U\in\mathbf{Cart}\),
\[
S^1(U)=\mathrm{C}^\infty(U,\mathbb{R})/\sim
\]
where \(f\sim g\) iff there exists a locally constant integer-valued function \(n:U\to\mathbb{Z}\) with \(g=f+n\). Restriction maps are given by restriction of representatives.
Local triviality
-
Local sections / epimorphism from \(h_{\mathbb R}\). The canonical projection
\[
\pi:h_{\mathbb{R}}\longrightarrow S^1
\]
is an epimorphism of sheaves (indeed a covering in the sheaf-theoretic sense): for any \(U\) and any section \(\sigma\in S^1(U)\) there is an open cover \(U=\bigcup_i U_i\) such that on each \(U_i\) the class \(\sigma|_{U_i}\) has a representative \(f_i\in\mathrm{C}^\infty(U_i,\mathbb{R})\). This is immediate from the definition: \(\sigma\) is represented locally by smooth lifts.
-
Locally isomorphic to \(h_{\mathbb R}\). For any real number \(t\in\mathbb{R}\) consider the open interval \(I_t=(t-\tfrac12,t+\tfrac12)\subset\mathbb{R}\). The restriction of \(\pi\) to \(h_{I_t}\subset h_{\mathbb R}\) is injective on sections (no two distinct maps \(U\to I_t\) differ by an integer). Hence the projection \(\pi\) admits local sections that identify \(S^1|_{I_t}\) with \(h_{I_t}\). In sheaf terms: there exist open subfunctors of \(S^1\) (images of these \(h_{I_t}\)) which cover \(S^1\) and each is representable by an open subset of \(\mathbb{R}\). Therefore \(S^1\) is an object of the sheaf topos that is locally representable by \(\mathbb{R}\) (i.e. an étale space with local charts given by open subsets of \(\mathbb{R}\)). This reproduces the usual local manifold charts for the circle.
-
Group structure. The group structure on \(\mathbb{R}\) (addition) descends to \(S^1\) because \(\underline{\mathbb{Z}}\) is a subgroup of translations: addition on \(h_{\mathbb R}\) is compatible with the \(\underline{\mathbb{Z}}\)-action, so \(S^1\) inherits a (commutative) group structure in \(\mathbf{Sh}(\mathbf{Cart})\). Thus \(S^1\) is a sheaf of groups.
-
Sections over test objects. For a test object \(U\) the elements \(S^1(U)\) are equivalence classes of smooth real-valued functions modulo locally constant integer shifts:
\[
S^1(U)=\mathrm{C}^\infty(U,\mathbb{R})/\{f\sim f+n\ (n\text{ locally constant }U\to\mathbb{Z})\}.
\]
In particular, for \(U=\mathbb{R}^n\) this exactly matches the intuitive notion of smooth maps \(\mathbb{R}^n \to S^1\) because locally one can lift a map to a real-valued function, and two lifts differ by integer-valued locally constant functions (the local monodromy).
-
Descent / sheaf condition. The equivalence relation by locally constant integers is local: if \(f\) and \(g\) restrict on a cover to functions differing by integers, then they differ by a locally constant integer function on the whole domain (glue the locally constant integers). Thus the presheaf quotient satisfies the sheaf condition, i.e. we have compatibility with covers.
Summary
We defined \(S^1\) to be the coequalizer sheaf
\[
S^1 := \operatorname{coeq}\big(\underline{\mathbb{Z}}\times h_{\mathbb{R}}
\rightrightarrows h_{\mathbb{R}}\big)
\]
with the obvious projection and translation action.
We also verified the following properties:
- For each test object \(U\), \(S^1(U)=\mathrm{C}^\infty(U,\mathbb{R})/\sim\) where \(f\sim g\) iff \(g=f+n\) for some locally constant \(n:U\to\mathbb{Z}\).
- \(S^1\) is a sheaf (the equivalence relation is local).
- The projection \(h_{\mathbb{R}}\to S^1\) is a covering epimorphism; \(S^1\) is locally representable by open intervals in \(\mathbb{R}\). Hence \(S^1\) is a 1-dimensional manifold object in the sheaf topos (i.e. locally isomorphic to \(h_{\mathbb{R}}\)).
- \(S^1\) inherits a commutative group object structure from \(\mathbb{R}\) modulo \(\underline{\mathbb{Z}}\).