nLab global element

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 $C$ has a terminal object $1$, a global element of another object $x$ is a morphism $1 \to x$.

So a global element is a generalized element at “stage of definition” $1$.

For example:

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

• In Cat, global elements are objects: the terminal category $1$ is the discrete category with one object, and a functor from $1$ into a category $C$ singles out an object of $C$.

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

• Working in a slice category $C/b$, a global element of the object $\pi: e \to b$ is a map into it from the terminal object $1_b: b \to b$; i.e., a right inverse for $\pi$. In the context of bundles, a global element of a bundle is called a global section.

If $C$ does not have a terminal object, we can still define a global element of $x\in C$ to be a global element of the represented presheaf $C(-,x) \in [C^{op},Set]$. Since the Yoneda embedding $x \mapsto C(-,x)$ is fully faithful and preserves any limits that exist in $C$, including terminal objects, if $C$ 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 $x$ consists of, for every object $i\in C$, a morphism $e_x : i\to x$ (i.e. a generalized element of $x$ at stage $i$), such that for any morphism $f:i\to j$ we have $e_j \circ f = e_i$.

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 $x$ can be called an element of $x$. For example, an element of an abelian group $x$ is a morphism from the group $\mathbf{Z}$ of integers to $x$, and of course this is equivalent to the usual notion of element of $x$. Here the adjective ‘global’ would not conform to the usage above since $\mathbf{Z}$ is not terminal, although we warn that some authors may call a map $\mathbf{Z} \to A$ a “global element” of $A$.

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

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

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

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

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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. ↩