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Previous forum page with exactly the same title: https://nforum.ncatlab.org/discussion/9582/homotopy-theory-faq/
Why are they not merged?
Thanks for all this. Just to say that I think eventually the question “What is homotopy theory?” should be the first in the list and should be added some lead-in sentences with an actual explanation, before the list of technical terms is being discussed.
Yes, thanks. Presumably “What is homotopy theory?” would be answered along the same lines as the Idea section of homotopy theory. “What is string theory?” at string theory FAQ and the Idea at string theory appear to be written at a similar level of sophistication. I guess it gives an opportunity to provide a second account.
Quotations like Barwick’s in the existing first section could go there:
I think of homotopy theory as an enrichment of the notion of equality, dedicated to the primacy of structure over properties. Simplistic and abstract though this idea is, it leads rapidly to a whole universe of nontrivial structures.
Yes, something along these lines would be good to add.
For the purpose of basic exposition I would go further and add something like the first lines here.
Next, I think it would be good to address the sizeable group of people who are mathematically sophisticated, just not in homotopy theory. There are people who appreciate concepts like derived categories, dgc-algraic geometry, but who black out at the term “homotopy theory”. This audience deserves a sentence on how homotopy theory is secretly behind all these niceties.
I can add something myself. But not right now.
I renamed the current questions “What is …” to “What objects does … study”, since it describes them much more accurately.
I added a new section “What is homotopy theory?”. Somebody who understands the mindset of outsiders should fill it in.
I am hesitant to do this because it is far too easy to descend into handwavy arguments that will still be incomprehensible to outsiders. Maybe a bunch of concrete examples would help.
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