Definition. Generalized element [math-0010]

Let $\mathcal{E}$ be a category, and let $A$ be an object of $\mathcal{E}$ . A generalized element of $A$ is simply a map in $\mathcal{E}$ with codomain $A$ .

A generalized element $x : S \to A$ is shape $S$ or $S$ -element of $A$ .

When $S = 1$ (the terminal) then $S$ -elements are called global elements.

Global elements can be very boring, for example in the category of groups.

The language of generalized elements is the internal language of the category. For example, let $\mathcal{E}$ be a category with finite products. An $S$ -element of $X \times Y$ consists of two component $x : S \to X, y : S \to Y$ and is denoted by $(x,y)$ , hence extended the notation of set-theoretic Cartesian product of global elements.