point

The **point** is what is shown here:

$\bullet$

As a category, we can interpret this as a category with a single object $\bullet$ and a single morphism (the identity morphism on the object, which is not shown in the picture since it is automatic). This can be generalised recursively to higher categories; as an $(n+1)$-category, the point consists of a single object $\bullet$ whose endomorphism $n$-category is the point (now understood as an $n$-category). Of course, to make this work, the point must be a symmetric monoidal $n$-category at each stage, but it is (in a unique way). In the limit, the point can even be understood as an $\infty$-category, with a unique $j$-morphism for each $j \geq 0$ (each of which is an identity for $j \gt 0$).

In the other direction, the point is a singleton set with a unique element $\bullet$. It can also be seen as a truth value that is true. It can even be understood as the $(-2)$-category.

As a topological space, the *point space* is the usual point from geometry: that which has no part. In more modern language, we might say that it has no *structure* —except that something exists. (So it is *not* empty!) This is consistent with the preceding paragraphs using the interpretation of a topological space as an $\infty$-groupoid. (But up to homotopy equivalence, any contractible space qualifies as a point.)

In all of the above, the point can be seen as a terminal object in an appropriate category (or $\infty$-category). However, you can also see it as a null object in a category of pointed objects. (Of course, it's always true that a terminal object $1$ in $C$ becomes a null object in $1/C$, but the dual argument also holds, so the question is which is the primary picture.)

But perhaps the point is best seen as the unique object in itself:

$\bullet = \{\bullet\} ,$

an equation that makes sense as a definition in the theory of ill-founded pure sets. Another possible definition (this time well-founded) in pure set theory is that the point is $\{\emptyset\}$, but this doesn't capture the picture that we get from higher category theory: the $(-1)$-category (truth value) of the $(-2)$-category (the point) is true (which is also the point), the $0$-category (set) of the true truth value is the singleton (which is also the point), the $1$-category (category) of the singleton (and *all* of its endofunctions!) is the terminal category (which is also the point), and so on. That is:

$\bullet \in \bullet \in \bullet \in \bullet \in \cdots .$

The term ‘point’ is often used for a global element; that is the meaning, for example, in the sense of a point of a topos or a point of a locale. The connection is that a global element of $X$ is a map from the point to $X$. So one may describe the point above as the *abstract* point, while a global element is a *concrete* point.

homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|

h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |

h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |

h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |

h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |

h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |

h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |

h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |

h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |