nLab
generalized element

In a category $C$, for any objects $c\in C$ and $I\in C$, the hom-set

$${\mathrm{Hom}}_C(I,c)$$Hom_C(I,c)

can be referred to as the set of generalized elements of $c$ with domain (or “stage of definition”) $I$.

Motivation

The primordial example is when $C$ is the category Set of sets and $I$ is a terminal object in $\mathrm{Set}$ — that is, a set with one element. Then elements of any set $c$ are in one-to-one correspondence with functions $f:I\to c$. This correspondence works as follows: given any element of $c$ there is a unique function $f:I\to c$ with this element in its image, and conversely each function $f:I\to c$ has a unique element of $c$ in its image.

In the same way, in a concrete category represented by $I$, the $I$-elements of an object are the same as the elements of its underlying set. (The category of sets is actually a special case of this, since it is concrete and represented by a terminal object.)

Generalizing from $\mathrm{Set}$ in another way, in any category with a terminal object $I$, we call a morphism $f:I\to c$ a global element of the object $c$. Generalizing further, it is common to take $I$ to be the unit object whenever $C$ is a monoidal category; in this case the generalized elements are important in enriched category theory). Note that any closed monoidal category is again a concrete category represented by this $I$.

The most general case where a single object $I$ may be used to define global elements is where $I$ is a generator of the category. However, not every category has a single object as a generator! Instead, in arbitrary categories, generalized elements of all possible stages of definition must often be used to replace global elements. Thus while a set is determined by its global elements, an object of an arbitrary category is determined by all of its generalized elements (this is one way to state the Yoneda lemma).

Examples

• For $C=$ Set and $I=\{•\}$ the singleton set, the global elements of a set $C$ are precisely the ordinary elements of $C$.

• For $C=[D^{\mathrm{op}},\mathrm{Set}]$ a presheaf category and for $I={\Delta }_{\mathrm{pt}}=(d\mapsto \{•\})$ the presheaf constant on the singleton set, the generalized elements of a presheaf $F$ are the global sections of this presheaf, equivalently these are the elements in the limit set over $F$.

• Again for $C=[D^{\mathrm{op}},\mathrm{Set}]$ and $I=D(-,d)$ a representable presheaf, the generalized elements of $F$ at stage $d$ are precisely the elements of the set $F(d)$, by the Yoneda lemma.

• An element in an abelian category is an equivalence class of generalised elements.