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I'm reading old papers on generalized recursion theory, and I've run across a paper by Van de Wiele ("Recursive dilators and generalized recursions") that 'lives in' the category ON, whose objects are ordinals and whose morphisms are strictly increasing maps. My question is, is there a good source to learn about the basic structure of this category? For instance, Van de Wiele writes: "In ON any ordinal is the direct limit of a system of integers" without proof, and - although I assume this is a fairly simple fact - I can't prove it.

Relevant background: I'm not a category theorist, so I'm looking for something fairly basic. I just want to understand ON well enough to follow Van de Wiele's paper.

Thanks!

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He's basically asserting that any ordinal is the supremum of a countable sequence of ordinals (which is false). –  Zhen Lin Jan 18 '13 at 21:07
    
Maybe he meant to write "countable ordinals." –  user28111 Jan 18 '13 at 21:53
    
Well, then that's obviously true: the cofinality of a countable ordinal is countable, and any cofinal $\omega$-sequence gives the required direct system. –  Zhen Lin Jan 18 '13 at 22:36
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1 Answer

up vote 2 down vote accepted

Let $\alpha$ be an ordinal. Let $S$ be the set of finite subsets of $\alpha$. Consider the following direct system:

  • For each $x\in S$, there is an object $a_x=|x|$.
  • For $x,y\in S$ with $x\subset y$, there is the corresponding morphism $f_{x,y}\colon a_x\to a_y$. That is if $x=\{x_0,\ldots,x_n\}$ and $y=\{y_0,\ldots,y_m\}$ (both in ascending order), then $f_{x,y}(i)=j$ iff $x_i=y_j$.

Matching this direct system, we have the morhisms $\phi_x\colon a_x\to \alpha$ given by $i\mapsto x_i$ (again assuming $x=\{x_0,\ldots,x_n\}$ in ascending order). The required property $\phi_x=\phi_y\circ f_{x,y}$ is clear.

I claim that $\alpha$ (together with the $\phi_x$) is a direct limit of the direct system specified above. Let $\beta$ be an ordinal together with morphisms $\psi_x\colon a_x\to b$ such that $\psi_x=\psi_y\circ f_{x,y}$ whenever $x\subset y$. Then we can define a morphism $h\colon \alpha\to \beta$ as follows: For $\gamma\in \alpha$, let $x\in S$ be a finite subset of $\alpha$ with $\gamma\in x$. Then $\gamma = \phi_x(i)$ for some $i\in a_x$. Let $h(\gamma)=\psi_x(i)$. This is well-defined for if also $\gamma \in y$, $\gamma=\phi_y(j)$ with $j\in a_y$, then also $\gamma \in x\cap y$ and via $f_{x\cap y,x}$ and $f_{x\cap y,y}$ we verify that $\psi_x(i)=\psi_y(j)$. Also, it is clear that we are forced to define $h$ like this, thus establishing the universal property of direct limit.

With hindsight, we could have done with the smaller set $$S=\alpha\cup \bigl\{\{\alpha_1,\alpha_2\}\mid \alpha_1<\alpha_2<\alpha\bigr\}. $$

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Ah, so what the author meant is a system whose vertices are sets of integers, rather than an integer-indexed system... –  Zhen Lin Jan 19 '13 at 21:56
    
That's what I rad into the expression "system of integers". –  Hagen von Eitzen Jan 19 '13 at 22:22
    
I'm still interested in sources on this category, but I'll accept this answer as it much clarified things for me. Thanks! –  user28111 Jan 27 '13 at 23:32
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