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 , can more generally be said and done for morphisms , for any other set . The point is just that many constructions can be performed “elementwise”. For instance, the fact that elements of are exactly pairs of an element of and an element of , when performed “elementwise” for morphisms out of , 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 .
On the other hand, once elements of objects are regarded as morphisms into these objects, the same reasoning applies to every category . Accordingly, for any category and an object of , one may refer to a morphism a generalized element of . One says this is a generalized element with stage of definition given by , or a figure of shape in .
The perspective of generalized elements of objects of a category is related to regarding as its image under the Yoneda embedding
that sends each object of to the set of generalized elements of at stage .
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 is the category Set of sets and is a terminal object in — that is, a set with one element. Then elements of any set are in one-to-one correspondence with functions . This correspondence works as follows: given any element of there is a unique function with this element in its image, and conversely each function has a unique element of in its image.
In the same way, in a concrete category whose underlying-set functor is represented by , the -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 in another way, in any category with a terminal object , we call a morphism a global element of the object .
Observe that in , an ordinary (i.e. global) point of an object , a section , corresponds to a generalized element in . Thus the slice theorem guarantees that generalized points with domain may be treated exactly as ordinary points, just in a more variable topos .
Arguably, the most general case where generalized elements defined at only one stage are “sufficient” when 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 a presheaf category and for the presheaf constant at the singleton set, the generalized elements of a presheaf are the global sections of this presheaf, equivalently these are the elements in the limit set over .
An element in an abelian category is an equivalence class of generalised elements.
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.