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What are examples of mathematical objects that are somehow 'constructed out of' a whole range of other objects but fall out of them? One example that comes to my mind is that of ordinal numbers: $\omega$ is constructed by putting together (i.e., the union of) all natural numbers $n$, but falls out of their realm, $\omega^2$ is constructed out of all ordinals $\omega+i$, where $i$ is a natural but falls out of them, etc. I guess the class of all sets is another example? It is, in a way, built of all sets yet falls out of their realm. On the other hand, natural numbers aren't like that.

Do you know of other examples, or where to look for more? (It doesn't have to be set-theoretic in nature.) How common is this phenomenon?

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    $\begingroup$ The whole notion of 'completion' (of a topological space, etc.) is to expand the range by this sort of process. $\endgroup$
    – Steven Stadnicki
    Commented May 14, 2021 at 18:53
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    $\begingroup$ There are a lot of categories whose objects are colimits of small objects. For example, any set is a filtered colimit of finite sets, any commutative ring is a filtered colimit of finitely presented commutative rings, any module over a ring is a colimit of finitely presented modules, etcetera. The keyword is accessible categories. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 14, 2021 at 18:57
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    $\begingroup$ I think $\omega^2$ should be built out of $\omega i+j$ for $i,j$ natural. $\endgroup$
    – Wojowu
    Commented May 14, 2021 at 18:59
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    $\begingroup$ Modular curves are often (but not always) curves of genus $\ne 1$. $\endgroup$
    – Steve Huntsman
    Commented May 14, 2021 at 20:09

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Consider addition on the natural numbers. Abstract algebra, as a field of mathematical study, is essentially born from the idea of taking that operation (and others) as mathematical objects in their own right.

Many (perhaps all?) mathematical fields are similarly born from objectifying some abstract concept.

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