# Timeline of "foundational" advances in homotopy theory?

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2022-06-23 07:00:08

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As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been a narrowing of focus, via a choice to ignore certain "pathological"—or at least intractably complicated—phenomena; instead of considering all topological spaces, one focuses only on compactly generated spaces or CW complexes or something. Or maybe one chooses to focus only on stable homotopy groups. Other times, there seems to have been a broadening of perspective, as new objects of study are introduced to fill perceived gaps in the landscape. Spectra are one notable example. I was fascinated when I discovered the 1991 paper by Lewis, Is there a convenient category of spectra?, showing that a certain list of seemingly desirable properties cannot be simultaneously satisfied. More recent concepts include model categories, $\infty$ -categories, and homotopy type theory.

I was wondering if someone could sketch a timeline of the most important such "foundational shifts" in homotopy theory over the past 50 years, together with a couple of brief sentences about what motivated the shifts. Such a bird's-eye sketch would, I think, help mathematicians from neighboring fields get some sense of the purpose of all the seemingly high-falutin' modern abstractions, and reduce the impenetrability of the current literature.