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. Here the adjective ‘global’ would not conform to the usage above since Z\mathbf{Z} is not terminal, although we warn that some authors may call a map ZA\mathbf{Z} \to A a “global element” of AA.

Thus generally, when CC is cartesian closed or even semicartesian monoidal closed, the monoidal unit II is terminal and such elements IAI \to A are global elements in the sense of this article. But again, even in the general monoidal case, some authors call maps of the form IAI \to A “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 𝒞\mathcal{C} be closed monoidal, with external (SetSet-valued) homs denoted by 𝒞(A,B)\mathcal{C}(A, B), the monoidal product by \otimes and the monoidal unit by II, and internal homs by [A,B][A, B]. Then for each pair (A,B)(A, B), the evident composite map

𝒞(A,B)𝒞(λ A,1 B)𝒞(IA,B)𝒞(I,[A,B])\mathcal{C}(A, B) \stackrel{\mathcal{C}(\lambda_A, 1_B)}{\to} \mathcal{C}(I \otimes A, B) \cong \mathcal{C}(I, [A, B])

(λ A:IAA\lambda_A: I \otimes A \to A the left unit isomorphism) defines a map which we denote as name A,B:𝒞(A,B)𝒞(I,[A,B])name_{A, B}: \mathcal{C}(A,B)\rightarrow\mathcal{C}(I, [A,B]). Notice this is the component at (A,B)(A, B) of a natural bijection namename; it takes a map f:ABf: A \to B in 𝒞\mathcal{C} to its name, typically denoted as "f":I[A,B]\text{"}f\text{"}: I \to [A, B], and which is an element of the internal hom [A,B][A, B].

Finally, 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.


  1. See, e.g., John Baez, Quantum Gravity Seminar, University of California, Riverside, Fall 2006, notes taken by Derek Wise, lecture of 2 November 2006.

Revised on July 17, 2017 12:14:47 by Todd Trimble (24.146.226.222)