definition. element [kostecki2011introduction, 2.8, 2.9] [tt-000M]

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Let \(X, S \in \operatorname {Ob}({\cal C})\).

An element or a generalized element of \(X\) at stage \(S\) (or, of shape \(S\)) is an arrow \(x : S \to X\) in \({\cal C}\), also denoted \(x \in _{S} X\).

An arrow \(1 \to X\) is called a global element of \(X\), a.k.a. a point of \(X\).

An arrow \(S \to X\), if \(S\) is not isomorphic to \(1\), is called the local element of \(X\) at stage \(S\).

An arrow \(\mathit {1}_X : X \to X\) is called the generic element of \(X\).