Sheaves are 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} $: $$ F: \text{Open}(X)^{\text{op}} \to \mathbf{Set} $$

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 $: $$ \rho_{U_i, U_i \cap U_j}(s_i) = \rho_{U_j, U_i \cap U_j}(s_j) $$ 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 $: $$ \rho_{U_i, U_i \cap U_j}(s_i) = \rho_{U_j, U_i \cap U_j}(s_j) $$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

Tags: algebraic-geometry category-theory descent-theory sheaves