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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?2012-11-03
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    A theorem of Freyd says that the closed unit interval $[0, 1]$ is the terminal object of a [certain category of coalgebras](http://ncatlab.org/nlab/show/coalgebra+of+the+real+interval).2012-11-03
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    @ Zhen Lin Thank you very much. That Dynamic-like object was exactly what I was hopping to see.2012-11-03
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    @ZhenLin Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-22
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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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