Definition:Compatible Family of Sections on Topological Space

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $\map {\mathbf {Ouv} } T$ be the category of open sets of $T$.

Let $\map {\mathbf {Ouv} } T ^{\mathrm {op} }$ be the dual category of $\map {\mathbf {Ouv} } T$.

Let $\FF : \map {\mathbf {Ouv} } T ^{\mathrm {op} } \to \mathbf {Set}$ be a presheaf of sets on $T$.

Let $U \subset S$ be an open subset of $T$.

Let $\family {U_i}_{i \mathop \in I}$ be an open cover of $U$.

Let $\family {f_i}_{i \mathop \in I}$ with $f_i \in \map \FF {U_i}$ for $i \mathop \in I$ be a family.

Let

${\operatorname {res}_{U_i \mathop \cap U_j}^{U_i}} : \map{\FF}{U_i} \to \map{\FF}{U_i \cap U_j}$

${\operatorname {res}_{U_j \mathop \cap U_j}^{U_j} } : \map{\FF}{U_j} \to \map{\FF}{U_i \cap U_j}$

for $i,j \mathop \in I$ be the restriction maps of the presheaf $\FF$.

Then $\family {f_i}_{i \mathop \in I}$ is a compatible family of sections if and only if

$\forall i, j \in I : \map {\operatorname {res}_{U_i \mathop \cap U_j}^{U_i} } {f_i} = \map {\operatorname {res}_{U_i \mathop \cap U_j}^{U_j} } {f_j}$

Also see

• Definition:Sheaf of Sets on Topological Space