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The paper Chromatic homotopy theory is asymptotically algebraic (by Barthel, Schlank and Stapleton) is an interesting example. In chromatic homotopy theory we study the category $\mathcal{L}(p,n)$ of spectral local with respect to the Johnson-Wilson theory $E(p,n)$, where $p$ is a prime and $n$ is a positive integer. It is a folklore idea that when $p$ is large relative to $n$, the properties of $\mathcal{L}(p,n)$ become more and more algebraic and more and more independent of $p$, in some sense. The cited paper formalises this idea by taking a kind of ultraproduct over primes. There is also a sequel called Monochromatic homotopy theory is asymptotically algebraic.