40 votes

Timeline of "foundational" advances in homotopy theory?

Such a timeline is necessarily highly subjective. With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory. The list below concentrates ...
34 votes
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Mark Hovey's open problems in the theory of model categories

I am a former student of Mark Hovey's, and during grad school, I wrote a document giving an update on the status of the 13 problems (as of 2012 or 2013, I guess). I just briefly went through it a ...
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24 votes

What happened to the last work Gaunce Lewis was doing when he died?

Lewis wrote, but never published, a very influential paper setting foundations for the multiplicative theory for Mackey functors. The paper is called "The Theory of Green functors" and Doug ...
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19 votes

Timeline of "foundational" advances in homotopy theory?

I think Dmitri Pavlov does a nice job laying out a timeline. Instead of writing a competing answer, let me just try to add in a couple of things I think he left out, addressing a few points from the ...
14 votes
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"Phantom" non-equivalences of spectra?

For this we can use a swindle-type technique. Let $B = \bigoplus_{n=2}^\infty H\Bbb Z/2$, and $A = \bigoplus_{n=2}^\infty \Sigma^{-n} H\Bbb Z/2$. We can construct maps $A \to B$ by specifying their ...
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10 votes

When did the Joyal model structure on simplicial sets originate?

My suspicion is now that it was some time between 2004 and 2006. I have a lot more citations in this blog post, but I note three points, in reverse chronological order: Multiple experts are referring ...
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8 votes

"Phantom" non-equivalences of spectra?

I am pretty sure that it will be rather delicate to find an example like you want. Here are some thoughts: Let $X_n$ denote $\Omega^{\infty} \Sigma^n X$. One is assuming an equivalence $X_n \simeq ...
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6 votes

Mark Hovey's open problems in the theory of model categories

On the other hand, the category of commutative monoids seems to be much more subtle. The conditions for the existence of a model structure on commutative monoids were worked out by Jacob Lurie around ...
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5 votes

Timeline of "foundational" advances in homotopy theory?

Thanks to both Pavlov and White there is now an almost complete list of "critical points" in the history of homotopy theory. There are a few items that perhaps should make it into the list, ...
5 votes
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Under which conditions is the bar construction a conservative functor?

$\vphantom{0pt}$ Hi Geoffroy. The natural way to approach this is by finding conditions under which the cobar functor preserves quasi-isomorphisms. If $A \to A'$ is a morphism of dg algebras, then you ...
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4 votes
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When is the Grothendieck / category of elements construction a fibration on geometric realizations?

Just so this question doesn't sit around forever with no "answer" (even though it's perfectly answered in the comments), I am writing a CW expansion to Cisinski's comment, to help folks who ...
4 votes
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On the proof of the surgery step in Wall's book

The theorem has the hypothesis "$f$ is in this class", meaning that the embedding $f$ is in the regular homotopy class of immersions determined by $F$ together with the element of the ...
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3 votes

Historical transition from classical homotopy to modern homotopy theory

This is not an answer, but a little too long for a comment. Scott, what a list: 3 dead, 2 retired, and me; also I think Mike and I are always on the same side of the fence. As to the original ...
3 votes

"Phantom" non-equivalences of spectra?

Here's a connective example. It is also an example of Maxime's variant question in the comments (regarding $\tau_{\leq m}$ truncations). And thanks to Maxime for looking this argument over before I ...
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1 vote

Space of algebraic maps, homotopy type of a CW complex

I find this question interesting, and I wish I had a stronger background in algebraic geometry to fully answer it. I had to look up the definition of algebraic maps and a remark reminded me that this ...
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1 vote

Operad terminology - Operads with and without O(0).

A 2020 paper Operads, monoids, monads, and bar constructions by Ruozi Zhang, Foling Zou, and J.P. May revisits the foundations of operad theory and discusses several possible choices for what $\...
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1 vote

Operad terminology - Operads with and without O(0).

EDIT: I did not realize there was still controversy, and I didn't know about the 2020 paper Peter May linked to today, but I like it a lot. I'm always happy to defer to Peter, so the terminology I'll ...
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