nelement

A nelement is a concept used in category theory to describe an “element” of an object in a way that does not rely on set membership. Formally, in any category with a terminal object 1, a nelement of an object A is a morphism e: 1 → A. The collection of nelements of A is the hom-set Hom(1, A). In this sense, nelements generalize the idea of elements from sets to arbitrary objects in a category.

In Set, there is a natural correspondence: since 1 is a singleton, a morphism e: 1 → A picks out a single element of A, so nelements correspond exactly to ordinary elements of A. In other categories, nelements may behave differently. For example, in the category of groups, there is still a unique morphism 1 → G, corresponding to the identity element, so the set of nelements can be small or even trivial compared to the underlying structure.

Nelements are functorial with respect to morphisms. If f: A → B is a morphism, composing with e ∈ Hom(1, A) yields f ∘ e ∈ Hom(1, B). This provides a natural way to transport elements along arrows and to study how subobjects and predicates interact internally to the category, particularly in a topos.

Usage and terminology vary. Some authors simply use “elements” or “points” for these morphisms, while others prefer nelements to emphasize the distinction from set-theoretic membership. See also element (category theory), global element, terminal object, and the Hom functor.