Could not include topos theory - contents
One striking difference between set theory and category theory is that, while objects of a category need not have any other structure, a set comes equipped with the notion of element, identifying other sets which belong to it. Sometimes, a category turns out to possess a similar structure, designating certain morphisms as global elements (or global points in geometric contexts) of an object.
If a category has a terminal object , a global element of another object is a morphism .
So a global element is a generalized element at “stage of definition” .
In Set, global elements are just elements: a function from a one-element set into picks out a single element of .
Working in a slice category , a global element of the object is a map into it from the terminal object ; i.e., a right inverse for . In the context of bundles, a global element of a bundle is called a global section.
Many (but not all) of the examples above are cartesian closed categories. In a more general closed category, a morphism from the unit object to can be called an element of . For example, an element of an abelian group is a morphism from the group of integers to , and of course this is equivalent to the usual notion of element of . But here the adjective ‘global’ is not used.
In contrast to a global element, a morphism to from any object whatsoever may be seen as a generalized element of . For example, if is the unit interval (in topology, chain complexes, etc), then a map from to is a path (rather than a point) in . Or in a slice category , if is an embedding, then a morphism from to is a local section of .