\begin{definition}[Definition:Sheaf of Sets on Topological Space]
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 $T$.
{{explain|$\map {\mathbf {Ouv} } T^{\mathrm {op} }$ 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.}}
\end{definition}