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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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    $\mathbb{R}$ 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.)2013-10-19

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

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    If you open a thread in meta and request *there* that I copy the comments there, I will. Here they are off topic — as you are well aware2017-03-03
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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.