nLab
global element

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Category Theory

Global elements

Idea

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.

Definition

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.

Variations

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 (80.92.246.195)