Points

The abstract 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:

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

Concrete points

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.

As homotopy types

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-$\infty$-groupoid