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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Started the page. First question: is homotopy theory a part of algebraic topology?

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020
    • (edited Apr 5th 2020)

    Previous forum page with exactly the same title: https://nforum.ncatlab.org/discussion/9582/homotopy-theory-faq/

    Why are they not merged?

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Added some possible further questions.

    Please feel free to answer!

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Added a section on homotopy theory.

    diff, v2, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Added a section on algebraic topology.

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2020
    • (edited Apr 5th 2020)

    Great idea!

    Have added hyperlinks to a few more keywords.

    diff, v4, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Added a section on “spaces”.

    diff, v5, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Added a section on models.

    diff, v5, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2020
    • (edited Apr 5th 2020)

    I have added redirects to a bunch of other entries in order to make hyperlinks here come out.

    (Not sure why the redirect to “Theta-n space” refuses to work.)

    I suggest that if you really want an entry “mathematical object” you start a stub right now.

    diff, v6, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Answered two more questions.

    diff, v7, current

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Answered another question.

    diff, v7, current

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Answered another question.

    diff, v7, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Answer the question about homotopical algebra.

    diff, v7, current

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Answer the question about model categories.

    diff, v7, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Answered the question about higher category theory.

    diff, v7, current

    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Answered the question about derived functors.

    diff, v7, current

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 6th 2020

    Finished answering questions.

    diff, v8, current

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 6th 2020

    Added references.

    diff, v9, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2020
    • (edited Apr 6th 2020)

    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.

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 6th 2020

    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.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2020

    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.

    • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 6th 2020

    Renamed two questions and added two new questions.

    diff, v10, current

    • CommentRowNumber23.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 6th 2020
    • (edited Apr 6th 2020)

    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.

    • CommentRowNumber24.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 8th 2020

    Answered the question about derived categories.

    diff, v11, current

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