generalized element

In a broad, non-technical sense, an “element” is a “building block”, “component”, or “basic part” of a more substantial whole. Ordinary or global elements of a set are simply the points of that set, and hence sufficiently capture this broad notion of “element” in Set, since by definition sets are no more than collections of points.

However, in general, knowing about the points of an object is insufficient to count as knowing its elements (construed broadly). From the point of view of the category Set, most things that can be said and done about elements of a set $X$, can more generally be said and done for morphisms $x \colon U\to X$, for *any* other set $U$. The point is just that many constructions can be performed “elementwise”. For instance, the fact that elements of $X\times Y$ are exactly pairs $(x,y)$ of an element of $X$ and an element of $Y$, when performed “elementwise” for morphisms out of $U$, expresses the universal property of a product. In structural set theory such as ETCS, one sometimes (but not necessarily) takes this point of view for axiomatizing the structure of $Set$.

On the other hand, once elements of objects are regarded as morphisms into these objects, the same reasoning applies to *every* category $C$. Accordingly, for $C$ any category and $X$ an object of $C$, one may refer to a morphism $x \colon U \to X$ a **generalized element** of $X$. One says this is a generalized element with **stage of definition** given by $U$, or a **figure** of shape $U$ in $X$.

The perspective of generalized elements of objects of a category $C$ is related to regarding $C$ as its image under the Yoneda embedding

$Y : C \hookrightarrow [C^{op}, Set]$

into its presheaf category. Under this embedding, every object $X$ of $C$ is mapped to the functor – the representable functor represented by it –

$GenEl(X) : C^{op} \to Set$

that sends each object $U$ of $C$ to the set of generalized elements of $X$ at stage $U$.

It is also worth noting that the internal logic or type theory of a category provides us a way to go backwards formally. By reasoning about “abstract elements” in a set-theoretic style like ordinary elements, the interpretation then “compiles” such proofs to category-theoretic ones which actually apply to all generalized elements.

The primordial example is when $C$ is the category Set of sets and $I$ is a terminal object in $Set$ — 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 whose underlying-set functor is 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, with the identity functor represented by a terminal object.)

Generalizing from $Set$ 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 $c$.

The stability or slice theorem for toposes says that if $\mathbf{E}$ is a topos, then also the slice category $\mathbf{E}/U$ is a topos for any object $U$, and the pullback functor $U^\ast: \mathbf{E} \to \mathbf{E}/U$ is a logical functor.

Observe that in $\mathbf{E}/U$, an ordinary (i.e. global) point of an object $U^\ast X$, a section $U^\ast 1 \to U^\ast X$, corresponds to a generalized element $U \to X$ in $\mathbf{E}$. Thus the slice theorem guarantees that generalized points with domain $U$ may be treated exactly as ordinary points, just in a more variable topos $\mathbf{E}/U$.

On the other hand, it is common to take $I$ to be the unit object whenever $C$ is a monoidal category. The generalized elements defined over this $I$ are important in enriched category theory).

Arguably, the most general case where generalized elements defined at only one stage $I$ are “sufficient” when $I$ is some sort of generator of the category. However, not every category has a single object as any sort of 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).

For $C = [D^{op}, Set]$ a presheaf category and for $I = \Delta_{pt} = (d \mapsto \{\bullet\})$ the presheaf constant at 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 $F$.

On the other hand, if $I=D(-,d)$ is a representable presheaf, then 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.

In the internal type theory of a category $C$, the generalized elements of $X$ at stage $U$ can be identified with terms of type $X$ in context $u\colon U$:

$u\colon U \vdash x(u) \colon X
\,.$

See the references below.

The fact that all type-theoretic constructions can be performed in any context implies that we can manipulate ordinary elements, and end up speaking also about generalized elements defined at arbitrary stages.

The interpretation of terms in type theory as generalized elements of objects in a category is discussed for instance on p. 8 of

- Steve Awodey, Andrej Bauer,
*Propositions as $[$Types$]$*, Journal of Logic and Computation. Volume 14, Issue 4, August 2004, pp. 447-471 (pdf)