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Our very first post here spoke about the Fraïssé amalgam, a way of constructing a universal object out of a countable set of similar objects satisfying certain conditions. For example the amalgam of the set of finite graphs is the infinite random graph.
Alexandre was asking there why the results of such amalgamation should be the kinds of entity we encounter through different routes. I should imagine that the answer to this has much in common with answers to the questions Michiel Hazewinkel is posing in Niceness Theorems:
Many things in mathematics seem almost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric…
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