topos theory

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 $C$ 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.

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 $Z$ of integers to $x$, and of course this is equivalent to the usual notion of elment of $x$. But here the adjective ‘global’ is not used.

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 $\pi$.

Revised on April 13, 2012 17:02:24 by Cameron Smith (129.98.105.162)