global element


Topos Theory

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Category Theory

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. 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 (