nLab
element

An element of a set is a thing which “belongs to,” or “is an element of,” that set.

The circularity of this definition is unavoidable in foundational set theories in which “set” is an undefined term. In “definitional” set theories, where “set” is defined in terms of something else, elements are likewise defined in terms of the same “something else.”

Generalisations

• An object in a category;
• a morphism in a category, a different sort of element from an object;
• in general, $k$-morphisms or $k$-cells in an $\mathrm{\infty }$-category;
• a generalised element of any object in any category or higher category;
• a global element of any object in any category or higher category with a terminal object.