Sheaves as Presheaves with an Equalizer Diagram

A sheaf on a topological space $X$ can be defined as a presheaf $F: \text{Open}(X)^{\text{op}} \to \mathbf{Set}$ such that for any open covering $\{ U_i \}$ $(i \in I)$ of an open set $U$, the following diagram is an equalizer:

Presheaves and sheaves

A presheaf $F$ on $X$ is a contravariant functor from the collection of open sets of $X$ ordered by inclusion, $\text{Open}(X)$, to the category of sets, $\mathbf{Set}$:

For every inclusion of open sets $V \subseteq U$, there is a restriction map $\rho_{U,V}: F(U) \to F(V)$.

A sheaf $F$ is a presheaf that satisfies two additional axioms (the sheaf axioms):

Locality (first sheaf axiom): If $\{ U_i \}$ $(i \in I)$ is an open covering of $U$ and $s, t \in F(U)$ are such that $\rho_{U,U_i}(s) = \rho_{U,U_i}(t)$ for all $i \in I$, then $s = t$.

Gluing (second sheaf axiom): If $\{ U_i \}$ $(i \in I)$ is an open covering of $U$ and $\{ s_i \in F(U_i) \}$ $(i \in I)$ is a family of sections such that for all $i, j \in I$:

then there exists a unique $s \in F(U)$ such that $\rho_{U,U_i}(s) = s_i$ for all $i \in I$.

The equalizer diagram

The sheaf condition can be expressed in terms of an equalizer diagram in the category $\mathbf{Set}$:

An equalizer of two parallel morphisms $f, g: A \to B$ in a category is an object $E$ and a morphism $e: E \to A$ such that $f \circ e = g \circ e$, and $e$ is universal with this property.

In the context of presheaves, for each open covering $\{ U_i \}$ of $U$, the sheaf condition can be formulated as the assertion that $F(U)$ is the equalizer of the following diagram:

a. The map $\delta: F(U) \to \prod_{i\in I} F(U_i)$

For a section $s \in F(U)$, $\delta(s)$ is the family $(\rho_{U,U_i}(s))$ $(i \in I)$. $\delta$ maps a global section over $U$ to its restrictions over the open cover $\{ U_i \}$.

b. The maps $\alpha, \beta: \prod_{i\in I} F(U_i) \to \prod_{(i,j) \in I \times I} F(U_i \cap U_j)$

For a family $(s_i)$ $(i \in I) \in \prod_{i\in I} F(U_i)$, $\alpha$ maps it to $(\rho_{U_i, U_i \cap U_j}(s_i))_{(i,j) \in I \times I}$.

Similarly, $\beta$ maps $(s_i)$ $(i \in I)$ to $(\rho_{U_j, U_i \cap U_j}(s_j))_{(i,j) \in I \times I}$.

$\alpha$ gives the restrictions of each $s_i$ to the overlaps $U_i \cap U_j$ and $\beta$ gives the restrictions of each $s_j$ (note the index $j$) to $U_i \cap U_j$.

c. The equalizer diagram

The diagram expresses that $F(U)$ consists exactly of those sections whose restrictions to $U_i$ match the given compatible family $(s_i)$.

The equalizer condition requires that $\alpha \circ \delta = \beta \circ \delta$ and that any $(s_i)$ satisfying $\alpha(s_i) = \beta(s_i)$ comes from a unique $s \in F(U)$.

We will show that the sheaf condition is equivalent to the diagram being an equalizer.

a. If $F$ is a sheaf, the diagram is an equalizer

First sheaf axiom (locality):

Suppose $s, t \in F(U)$ have the same image under $\delta$, i.e., $\delta(s) = \delta(t)$. Then $\rho_{U,U_i}(s) = \rho_{U,U_i}(t)$ for all $i \in I$. By the first sheaf axiom, this implies $s = t$. So, the map $\delta$ is monic (injective), meaning $F(U)$ embeds into $\prod_{i\in I} F(U_i)$.

Second sheaf axiom (gluing):

Suppose $(s_i)$ $(i \in I) \in \prod_{i\in I} F(U_i)$ satisfies $\alpha(s_i) = \beta(s_i)$, i.e., for all $i, j$:

Then the second sheaf axiom ensures there exists a unique $s \in F(U)$ such that $\rho_{U,U_i}(s) = s_i$ for all $i \in I$. So, every compatible family $(s_i)$ arises from a unique $s \in F(U)$, so $F(U)$ is the equalizer of $\alpha$ and $\beta$.

b. If the diagram is an equalizer, $F$ is a sheaf

First sheaf axiom (locality): If $\rho_{U,U_i}(s) = \rho_{U,U_i}(t)$ for all $i \in I$, then $\delta(s) = \delta(t)$. Since $\delta$ is monic (as an equalizer), $s = t$.

Second sheaf axiom (gluing):

Suppose we have $(s_i)$ $(i \in I)$ such that $\rho_{U_i, U_i \cap U_j}(s_i) = \rho_{U_j, U_i \cap U_j}(s_j)$ for all $i, j$. This means $\alpha(s_i) = \beta(s_i)$. Since $F(U)$ is the equalizer, there exists a unique $s \in F(U)$ such that $\delta(s) = (s_i)$, i.e., $\rho_{U,U_i}(s) = s_i$.

Read more: Facets of descent, I George Janelidze