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” .
For example:
In Set, global elements are just elements: a function from a one-element set into picks out a single element of .
In Cat, global elements are objects: the terminal category is the discrete category with one object, and a functor from into a category singles out an object of .
In a topos, a global element of the subobject classifier is called a truth value.
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.
If does not have a terminal object, we can still define a global element of to be a global element of the represented presheaf . Since the Yoneda embedding is fully faithful and preserves any limits that exist in , including terminal objects, if does have a terminal object then this definition coincides with the more naive one.
If we unravel the general definition more explicitly, it says that a global element of consists of, for every object , a morphism (i.e. a generalized element of at stage ), such that for any morphism we have .
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 . Here the adjective ‘global’ would not conform to the usage above since is not terminal, although we warn that some authors may call a map a “global element” of .
Thus generally, when is cartesian closed or even semicartesian monoidal closed, the monoidal unit is terminal and such elements are global elements in the sense of this article. But again, even in the general monoidal case, some authors call maps of the form “global elements”, even though this conflicts with our usage.
As an important special case, there is1 for closed monoidal categories a notion of “name of a morphism”, as follows. Let be closed monoidal, with external (-valued) homs denoted by , the monoidal product by and the monoidal unit by , and internal homs by . Then for each pair , the evident composite map
( the left unit isomorphism) defines a map which we denote as . Notice this is the component at of a natural bijection ; it takes a map in to its name, typically denoted as , and which is an element of the internal hom .
Finally, 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 .
See, e.g., John Baez, Quantum Gravity Seminar, University of California, Riverside, Fall 2006, notes taken by Derek Wise, lecture of 2 November 2006. ↩
Last revised on August 13, 2023 at 10:36:31. See the history of this page for a list of all contributions to it.