remark. element [kostecki2011introduction, 2.8] [tt-002W]

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In an element \(x : S \to X\), the object \(S\) is called a stage in order to express the intuition that it is a "place of view" on \(X\). In the same sense, \(S\) is also called a domain of variation, and \(X\) a variable element.

Sometimes, the term shape is used instead [leinster2016basic, 4.1.25], intuitive examples are:

• when the object is a set, a generalized element of \(X\) of shape \(\mathbb N\) is a sequence in the set \(X\)
• when the object is a topological space, a generalized element of \(X\) of shape \(S^1\) is a loop

In the context of studying solutions to polynomial equations, we may also call it a \(S\)-valued point in \(X\), where \(S\) is the number set where the solution is taken, e.g. the real, complex, and \(\operatorname {Spec} \mathbb F_p\)-valued solutions.