global element

Global elements


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 CC has a terminal object 11, a global element of another object xx is a morphism 1x1 \to x.

So a global element is a generalized element at “stage of definition” 11.

For example:

  • In Set, global elements are just elements: a function from a one-element set into xx picks out a single element of xx.

  • In Cat, global elements are objects: the terminal category 11 is the discrete category with one object, and a functor from 11 into a category CC singles out an object of CC.

  • In a topos, a global element of the subobject classifier is called a truth value.

  • Working in a slice category C/bC/b, a global element of the object π:eb\pi: e \to b is a map into it from the terminal object 1 b:bb1_b: b \to b; i.e., a right inverse for π\pi. 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 xx can be called an element of xx. For example, an element of an abelian group xx is a morphism from the group Z\mathbf{Z} of integers to xx, and of course this is equivalent to the usual notion of element of xx. But here the adjective ‘global’ is not used.

In contrast to a global element, a morphism to xx from any object ii whatsoever may be seen as a generalized element of xx. For example, if ii is the unit interval (in topology, chain complexes, etc), then a map from ii to xx is a path (rather than a point) in xx. Or in a slice category C/bC/b, if ρ:ab\rho: a \to b is an embedding, then a morphism from ρ\rho to π\pi is a local section of π\pi.

Revised on March 28, 2015 16:20:10 by Urs Schreiber (