Why is an arrow going out from a terminal object in natural number definition on nLab

category-theory

I have just started studying category theory.
In section 2 of the nLab page on "natural number objects", it talks about a morphism $z: 1 \rightarrow \mathbb{N}$. But it calls the $1$ object a terminal object! But shouldn't $1$ be called an initial object since there is one arrow going out from it to $\mathbb{N}$?

Best Answer

Recall the definition,

An object $t$ is said to be terminal if given any object $c$, there is a unique morphism $c \to t$

In $\mathsf{Set}$, any one point set is terminal. This is because given any set $A$ there is only one function $A \to \ast$, namely the constant function. Now, a pointed set consists of a set $A$ with a distinguished element $a \in A$. This can be thought of as a set $A$ and a function $* \to A$ which sends $*$ to $a$. The same can be done for many other examples, such as pointed spaces.

Thus, we can define a generalized notion of 'pointed objects'. Given a category $\mathcal{C}$ with a terminal object which we will call $1$, we define its category of pointed objects $\mathcal{C}_*$ (I can't recall if this is usual notation) as follows,

  • The objects are pairs $(c,a)$ with $c$ an object of $\mathcal{C}$ and, $a : 1 \to c$ an arrow from the terminal object to $c$. Note the resemblance with the example of sets.

  • The morphisms from $(c,a)$ to $(d,b)$ are arrows $f : c\to d$ such that "$f$ sends the point $a$ to the point $b$". This is formalized by requesting that $b = f \circ a$.

I don't know the context of the article (the link is broken for me as of now), but it may be talking about representing a selection of a natural number $n \in \mathbb{N}$ as an arrow $1 \to \mathbb{N}$ where $1$ is any terminal object (as two terminal objects are isomorphic).

Related Question