Sheaf Condition

2025-12-26

Defines Sheaf Condition, descent condition

Let $(\mathcal{C}, \tau)$ be a site and let $F: \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ be a presheaf.

Definition. The presheaf $F$ satisfies the sheaf condition (or descent condition) for a covering family $\{U_i \to U\}_{i \in I}$ in $\tau$ if the following diagram is an equalizer:

F(U) \xrightarrow{\;\prod_i \mathrm{res}_{U_i}\;} \prod_{i \in I} F(U_i) \rightrightarrows \prod_{i,j \in I} F(U_i \times_U U_j)

where the two parallel arrows send $(s_i)_i$ to the families $(\mathrm{res}(s_i))_{i,j}$ and $(\mathrm{res}(s_j))_{i,j}$ respectively. Concretely, this means:

(Gluing/Existence.) For every compatible family $(s_i \in F(U_i))_{i \in I}$ — meaning $s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j}$ for all $i, j$ — there exists a section $s \in F(U)$ with $s|_{U_i} = s_i$ for all $i$ .
(Separation/Uniqueness.) Such a section $s$ is unique: if $s, s' \in F(U)$ satisfy $s|_{U_i} = s'|_{U_i}$ for all $i$ , then $s = s'$ .

A sheaf on $(\mathcal{C}, \tau)$ is a presheaf satisfying the sheaf condition for every covering family in $\tau$ . A presheaf satisfying only condition (2) is called separated.

Proposition. The sheaf condition for $F$ with respect to a covering sieve $S \hookrightarrow \mathrm{Hom}(-, U)$ is equivalent to requiring that the induced map $\mathrm{Hom}(\mathrm{Hom}(-, U), F) \to \mathrm{Hom}(S, F)$ is an isomorphism (using the Yoneda embedding). That is, $F$ is a sheaf if and only if it is a local object with respect to all covering sieve inclusions.

Proposition. For sheaves of spaces in the $(\infty,1)$ -categorical setting, the sheaf condition is strengthened to homotopy descent: $F(U)$ must be equivalent to the homotopy limit

F(U) \xrightarrow{\;\sim\;} \mathrm{holim}\left(\prod_i F(U_i) \rightrightarrows \prod_{i,j} F(U_i \times_U U_j) \mathrel{\substack{\to \\ \to \\ \to}} \cdots \right)

of the full Cech cosimplicial diagram. Uniqueness of gluing is replaced by contractibility of the space of gluings.

Examples.

Topological sheaves: On the site $\mathrm{Open}(X)$ of a topological space $X$ with the open-cover topology, the sheaf condition is the classical gluing axiom: sections that agree on overlaps glue uniquely.
Representable presheaves: On any subcanonical site $(\mathcal{C}, \tau)$ , every representable presheaf $\mathrm{Hom}(-, C)$ is a sheaf.
Non-example: The presheaf of bounded functions on $\mathbb{R}$ (with the open-cover topology) is separated but not a sheaf: compatible bounded local sections glue to an unbounded global section when the local bounds grow without bound.

Sheaf Condition

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