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Natural numbers can be defined as the initial object of the category of pointed dynamical systems with triple $\left(X, s_0, f\right)$ where $f:X \rightarrow X$ and $s_o \in X$, as objects and conjugacy of dynamical systems as morphisms, i.e. a morphism $$\alpha: \left(X, s_0, f\right) \longrightarrow \left(Y, t_0, g\right)$$ satisfy $\alpha\circ f =g \circ \alpha$ and $\alpha \left(s_0\right)=t_0$.

Is it possible to enrich the following category, in order to be able to define the real numbers, as an initial object? I will be equally content to see how we can define computable numbers as some initial object. Many thanks.

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Fields with a complete ordered field embedded? – Cameron Buie Nov 3 '12 at 21:23
A theorem of Freyd says that the closed unit interval $[0, 1]$ is the terminal object of a certain category of coalgebras. – Zhen Lin Nov 3 '12 at 22:12
@ Zhen Lin Thank you very much. That Dynamic-like object was exactly what I was hopping to see. – Hooman Nov 3 '12 at 22:21
@ZhenLin Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. – Julian Kuelshammer Jun 22 '13 at 8:44
$\mathbb{R}$ is the initial object of the category of $\mathbb{R}$-algebras. (With this trivial comment I would like to indicate that the question is not precise enough.) – Martin Brandenburg Oct 19 '13 at 19:43

The field of real numbers is initial in the category of complete ordered fields.

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You could also view it is the initial object in the category with exactly one object, the field of the real numbers, and one morphism, the identity map. This sort of shows that your question needs some tweaking to be turned into an interesting one :-) – Mariano Suárez-Alvarez Oct 20 '13 at 0:03

Not sure if this is what you're looking for, but in the category of real Lie groups equipped with a tangent vector at the origin, $\mathbf R$ is the initial object: given a real Lie group $G$ and a tangent vector $v$ at the origin, there is a unique morphism of Lie groups $\mathbf R \to G$ which takes the unit tangent vector of $\mathbf R$ to $v$ (the exponential map). This is the "continuous" analogue of the fact that $\mathbf Z$, with its distinguished generator $1$, is the initial object in the category of pointed groups.

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