Definition:Sheaf of Sets 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 $\mathbf {Set}$ be the category of sets.

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

This article, or a section of it, needs explaining. In particular: $\map {\mathbf {Ouv} } T^{\mathrm {op} }$ and $\mathbf {Set}$. There is a lot of what appears to be category-theory specific notation going on here. Are these definitions genuinely part of the discipline of category theory? If not, then it is better to use more conventional language so that less well educated mathematicians have a chance of learning. i added explanations. The notation is conventional in category theory. Maybe one should also define Ouv(T) in category of open sets --Wandynsky (talk) 09:55, 28 July 2021 (UTC) if you prefer english notations, one can use Op(T) instead of Ouv(T). a) Whatever is standard in the (American) English mathematical community. This is an English language website. We must of course document the alternative notations in an "also denoted as" or "also known as" section, in the standard way of ProofWiki. This is what I mean when I suggest that starting at the end and working backwards is a suboptimal technique for populating this website, and leads to confusion and inconsistency. b) Is it really necessary to bring in the language of category theory? Is Sheaf Theory a subfield of category theory? I suspect the answer is no, in which case we need to either remove the category-theoretical notation from here or to sideline it. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Definition 1

$\FF$ is a sheaf of sets on $T$, if and only if it satisfies the following gluing property:

For any open subset $U \subset S$ of $T$ and

for any open cover $\family {U_i}_{i \mathop \in I}$ of $U$

and for any compatible family of sections $\family {f_i}_{i \mathop \in I}$ with $f_i \in \map\FF{U_i}$ for $i \mathop \in I$

there exists a unique $f \mathop \in \map \FF U$, such that

$\forall i \in I: \map {\operatorname {res}_{U_i}^U} f = f_i$

Definition 2

Let $\map {\operatorname {Sp\acute e} } \FF$ be the étalé space of $\FF$.

Let $\FF'$ be the sheaf of sections of $\map {\operatorname {Sp \acute e} } \FF \to T$.

$\FF$ is a sheaf of sets on $T$ if and only if the canonical mapping $\FF \to \FF'$ is an isomorphism.

Also see

• Equivalence of Definitions of Sheaf of Sets
• Sheaf of Sets iff Set-Valued Sheaf
• Definition:Sheaf on Topological Space