In the context of sheaf topos theory and differential geometry, it is crucial to establish that the simple spaces used to define generalized geometries are themselves sheaves. This property — where every representable presheaf is a sheaf — is known as being subcanonical.

We will show that the standard site of Cartesian spaces equipped with the open cover topology is subcanonical.

The Site of Cartesian Spaces ($\mathrm{CartSp}$)
Let $\mathcal{C} = \mathrm{CartSp}$ be the category where:
The Coverage
We equip $\mathrm{CartSp}$ with the standard Grothendieck topology. A covering families of an object $U \in \mathrm{CartSp}$ is a collection of morphisms $$ \mathcal{U} = \{ \iota_i : U_i \to U \}_{i \in I} $$ such that each $\iota_i$ is a smooth diffeomorphism onto an open subset of $U$, and the union of their images is the entirety of $U$: $$ \bigcup_{i \in I} \iota_i(U_i) = U. $$
Representable Presheaves
For any object $X \in \mathrm{CartSp}$, the representable presheaf associated with $X$ is the contravariant functor: $$ y(X) := \mathrm{Hom}_{\mathrm{CartSp}}(-, X) $$ For any probe space $U$, the set of plots is $y(X)(U) = C^\infty(U, X)$.
Theorem: The topology on $\mathrm{CartSp}$ is subcanonical.
That is, for every $X \in \mathrm{CartSp}$, the representable presheaf $y(X)$ is a sheaf.

To prove that $y(X)$ is a sheaf, we must verify the sheaf condition for an arbitrary cover. Let $U$ be a Cartesian space and let $\{ \iota_i : U_i \to U \}_{i \in I}$ be an open cover.

We examine the equalizer diagram in the category of Sets:

$$ C^\infty(U, X) \xrightarrow{\;e\;} \prod_{i \in I} C^\infty(U_i, X) \overset{r_1}{\underset{r_2}{\rightrightarrows}} \prod_{(i,j) \in I \times I} C^\infty(U_i \cap U_j, X) $$

Before proceeding, we define the maps involved:

  1. $e$ (restriction to a cover):
    This map takes a global smooth function $f$ and restricts it to every element of the cover. $$ e(f) = \{ f \circ \iota_i \}_{i \in I} $$ We also write $f\circ\iota_i = f_i = f|_{U_i}$.

  2. $r_1$ and $r_2$ (restriction to intersections):
    Let $\{f_k\}_{k \in I}$ be an element of the product space $\prod C^\infty(U_k, X)$. We compare restrictions on the intersections $U_{ij} = U_i \cap U_j$.
    • The map $r_1$ restricts the function indexed by the first component of the intersection pair: $$ \left( r_1(\{f_k\}) \right)_{ij} = f_i \big|_{U_i \cap U_j} $$
    • The map $r_2$ restricts the function indexed by the second component of the intersection pair: $$ \left( r_2(\{f_k\}) \right)_{ij} = f_j \big|_{U_i \cap U_j} $$

The sheaf condition states that $C^\infty(U, X)$ is the equalizer of $r_1$ and $r_2$. This requires proving two properties: Separatedness (Injectivity) and Gluing (Surjectivity onto the kernel).

Uniqueness (Separatedness)

Suppose we have two global maps $f, g \in C^\infty(U, X)$ such that $e(f) = e(g)$. This means:

$$ f|_{U_i} = g|_{U_i} \quad \text{for all } i \in I. $$

Since $\{U_i\}$ is a cover, for every point $p \in U$, there exists some index $k$ such that $p \in U_k$. Thus:

$$ f(p) = f|_{U_k}(p) = g|_{U_k}(p) = g(p). $$

Since the functions agree at every point $p$, $f = g$. Thus, the map $e$ is injective.

Existence (Gluing)

Suppose we are given a "matching family" of local sections. That is, a collection $\{f_i\}_{i \in I}$ where $f_i \in C^\infty(U_i, X)$ such that $r_1(\{f_i\}) = r_2(\{f_i\})$. Explicitly, this means they satisfy the compatibility condition:

$$ f_i \big|_{U_i \cap U_j} = f_j \big|_{U_i \cap U_j} \quad \text{for all pairs } (i,j). $$

We must construct a global map $f: U \to X$ such that $f|_{U_i} = f_i$.

Set-theoretic construction:
For any $p \in U$, choose an index $i$ such that $p \in U_i$. Define $f(p) = f_i(p)$. This is well-defined because if $p \in U_j$ as well, the compatibility condition ensures $f_i(p) = f_j(p)$.

Smoothness verification:
We must show that the constructed function $f$ is a morphism in $\mathrm{CartSp}$, i.e., it is smooth ($C^\infty$).

Smoothness is a local property. A function $f: U \to \mathbb{R}^n$ is smooth if and only if there exists an open cover of $U$ such that the restriction of $f$ to each set in the cover is smooth.

Therefore, $f$ is smooth globally.

Thus, for every matching family, there exists a unique global gluing $f$. The equalizer condition holds.

Q.E.D.