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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 1st 2020

redirect

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeMay 2nd 2020

He is now one of the main advocates of higher category theory in the United States.

really? Is this meant to mean the “most senior advocate”? As opposed to Lurie etc? May has even poured scorn over on MO on people who jump to model-independent $\infty$-categorical explanations.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 2nd 2020

Maybe “of low-dimensional higher category theory”, as per his more recent praise of 2-category theory (can’t find the link now, though).

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeMay 2nd 2020
• (edited May 2nd 2020)

I’d be interested to see that.

I seem to recall Lurie saying he didn’t really use 2-category theory, despite $(\infty, 2)$-Categories and the Goodwillie Calculus.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 2nd 2020

Found it: slide 2 of

• Peter May, Input for derived algebraic geometry: equivariant multiplicative infinite loop space theory Banff, February 18, 2016 (pdf)

announces that:

2-category theory ROCKS

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeMay 2nd 2020

Thanks. And here is what I was mentioning by Lurie:

I don’t think I’d consider that an honest application as of yet: the formalism of (infty,2)-categories is only used at the end to restate the earlier results, not to prove or improve upon them. However, I believe that the formalism of (infty,2)-categories will be extremely useful for getting deeper into the Goodwillie calculus (studying higher derivatives, understanding the chain rule, and so forth).

Another endorsement then.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 2nd 2020

To come back to the entry on Peter May:

I agree with David R. that the statement

He is now one of the main advocates of higher category theory in the United States.

seems dubious. Maybe the modification as I suggested

He is now one of the main advocates of 2-category theory…

is getting closer to the truth. But do we have more than one enthusistic remark in one public talk four years ago?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 4th 2020

This will be my last contribution here, but just for completeness:

• The line in question in #2 was added in revision 3, way back in June 2009. This was around May co-editing “Towards Higher Categories

• An example of the “scorn” on $\infty$-category theory mentioned in #2 in here.

I won’t edit the entry. But my suggestion is: Best not to claim what anyone is an “advocate” of, but instead to give primary sources of their writing: This book and that MO comment should really be linked to in the entry.

1. remove hilariously false claim

Anonymous

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeMay 4th 2020

Re #9, a particularly annoying tone! If you’d bothered to read the thread, it was in the process of being altered or removed.

In that Peter May had been one of the two leads on the summer school ’$n$-Categories: Foundations and Applications’ at the University of Minnesota (Minneapolis,7-18 June 2004) and editor of the book Urs mentions, it wasn’t ’hilariously false’. The meaning of ’higher category theory’ changed.

Try and contribute in a more collegial spirit.

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeMay 4th 2020

@David this person may not have been aware of the discussion thread, and just edited the page directly (for instance, I would not be a bit surprised if May himself did that edit, being alerted to the existence of the statement by a third party).

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeMay 5th 2020

Yes, perhaps. The timing is quite a coincidence though, unless something external prompted you to write #2.

• CommentRowNumber13.
• CommentAuthorDavidRoberts
• CommentTimeMay 5th 2020

No, just looking at the page after Dmitri’s edit. I hadn’t seen the contentious claim before.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeMay 5th 2020
• (edited May 5th 2020)

I had had the same thought as #11, but checked that the source IP seems to have been from Belgium.

And not to belittle politeness of words, I’ll say that it’s decent to take the responsibility of acting where action is clearly called for.

Therefore I have now contradicted my own plea of #8 and did edit the entry on the issue here, after all. Here is what I added. Hopefully this inspires somebody closer to Peter May to chime in and expand/improve:

Peter May’s work makes extensive use of enriched- and model-category theory as power tools in algebraic topology, notably in discussion of highly structured spectra in MMSS00’s Model categories of diagram spectra (for exposition see Introduction to Stable homotopy theory – 1-2), or in the discussion of K-theory of permutative categories. While he has co-edited a book collection on higher category theory (Baez-May 10) and eventually had high praise (May 16) for 2-category theory as a tool in algebraic topology/higher algebra, he has vocally warned against seeing abstract (∞,1)-category theory as a replacement of concrete computations in model category-theory (P. May, MO comment Dec 2013).

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeMay 5th 2020

@Urs

that’s excellent, thanks.

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeMay 5th 2020

Politeness is free. Anyway, something good came out of it.

• CommentRowNumber17.
• CommentAuthorMike Shulman
• CommentTimeMay 19th 2020

Coming to this thread a little late, the current version looks pretty good to me.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJan 11th 2021

added the references on ring spectra to the list of “Selected writings”