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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2010

    I worked on brushing up (infinity,1)-category a little

    • mostly I added in a section on homotopical categories, using some paragraphs from Andre Joyal's message to the CatTheory mailing list.

    • in this context I also rearranged the order of the subsections

    • I removed in the introduction the link to the page "Why (oo,1)-categories" and instead expanded the Idea section a bit.

    • added a paragraph to the beginning of the subsection on model categories

    • added the new Dugger/Spivak references on the relation between quasi-cats and SSet-cats (added that also to quasi-category and to relation between quasi-categories and simplicial categories)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 28th 2012

    Looking back at (infinity,1)-category I found that lots of context was missing there.

    As a first step in an attempt to correct this, I created a subsection “Properties” with some pointers to relevant other entries.

    • CommentRowNumber3.
    • CommentAuthoradeelkh
    • CommentTimeMay 14th 2013
    • (edited May 14th 2013)

    I added the reference

    • Omar Antolín Camarena, A whirlwind tour of the world of $(\infty,1)$-categories (arXiv)

    This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeAug 12th 2013
    • (edited Aug 12th 2013)

    In the entry on (infinity,1)-category there is the phrase:an (∞,1)-category is an internal to in ∞-groupoids/basic homotopy theory.

    I tried to see how to clear up the grammar, but it was not clear to me what the wording was intended to be. There was a previous version:

    To some extent an (∞,1)-category can be thought of as a category enriched in (∞,0)-categories, namely in ∞-groupoids.

    That is vague, so needed changing, but there seem to be ’typos’ in the current version.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeAug 12th 2013

    It was probably supposed to be “an internal category in …” or “a category internal to …”.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeAug 12th 2013
    • (edited Aug 12th 2013)

    Yes, but is that completely correct? The ‘enriched’ version to me was clearer. Is an (∞,1)-category really an internal category in ∞-groupoids, as that would mean the object of objects would be an ∞-groupoid, or am I mistaken?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2013
    • (edited Aug 12th 2013)

    Thanks for catching that, I have fixed the sentence now and expanded it such as to read as follows:

    More precisely, this is the notion of category up to coherent homotopy: an (,1)(\infty,1)-category is equivalently

    Is an (∞,1)-category really an internal category in ∞-groupoids, as that would mean the object of objects would be an ∞-groupoid,

    Yes, it’s the completeness condition of complete Segal spaces that takes care of this issue. Details are at internal category in an (∞,1)-category.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2015

    Added pointer to the new preprint by Ayala and Rozenblyum. Though it doesn’t seem to have the previously announced statement about (,n)(\infty,n)-categories with duals yet.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 7th 2015

    Has anything more been made of the other approach to stratified spaces where one moves up through strata and back down again? You may remember that discussion at the Cafe here. It gave rise to Transversal homotopy theory by Jon Woolf, who also wrote a paper mentioned by Ayala and Rozenblyum, The fundamental category of a stratified space.

    The idea was to give fundamental categories with duals.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2016
    • (edited Jan 19th 2016)

    have added pointer to

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 24th 2018

    Added reference to Riehl-Verity’s book.

    diff, v73, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2018
    • (edited Sep 12th 2018)

    added pointer to

    diff, v76, current

    • CommentRowNumber13.
    • CommentAuthorAli Caglayan
    • CommentTimeApr 3rd 2019


    diff, v77, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2019

    The pdf link in

    is broken, and (I seem to remember) the project was renamed, too. So I am removing this line hereby. But if you know which pointer should go here instead, please add it.

    diff, v78, current

    • CommentRowNumber15.
    • CommentAuthorjweinberger
    • CommentTimeMay 20th 2019

    Added the updated reference:

    diff, v79, current

    • CommentRowNumber16.
    • CommentAuthorTim_Porter
    • CommentTimeMay 20th 2019
    • (edited May 20th 2019)

    There is an old version on Doug Ravenel’s webpages: but I do not know its status.

    On Emily’s home pages is a reference and link to which is probably the best source for the new version. There is already a link to this on the n-Lab page.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2019

    Added the updated reference:


    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2020

    added publication data to

    diff, v80, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2020
    • (edited Jul 9th 2020)

    added publication data for this item in the list of references:

    What’s the latest that one could cite regarding survey of the web of Quillen equivalences between the different models for (,1)(\infty,1)-categories?

    Let’s see…

    Here is one, sort of:

    diff, v81, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2021

    added this pointer:

    diff, v83, current

    • CommentRowNumber21.
    • CommentAuthorEmily Riehl
    • CommentTimeAug 31st 2021

    Updated website for Elements of \infty-Category Theory

    diff, v88, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2021
    • (edited Aug 31st 2021)

    Fixed the link (it needs the https:// prefix in order to point outside of the domain

    I have also updated links accordingly at infinity-cosmos, at homotopy 2-category of quasi-categories and at Emily Riehl.

    diff, v89, current

    • CommentRowNumber23.
    • CommentAuthorMarc
    • CommentTimeSep 16th 2021

    repaired outdated link to Moritz Groth A short course on \infty-categories

    diff, v90, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2022
    • (edited May 8th 2022)

    added publication data and hyperlink to:

    diff, v95, current

    • CommentRowNumber25.
    • CommentAuthorHurkyl
    • CommentTimeOct 3rd 2022

    Changed “full sub-quasicategory” to “wide sub-quasicategory”in the definition of “relative quasicategory”.

    diff, v96, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2022
    • (edited Dec 8th 2022)

    added pointer to:

    • Emily Riehl, \infty-Category theory for undergraduates, talk at CQTS (Dec. 2022) [video: rec]

    diff, v97, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2023