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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011
    • (edited May 30th 2011)

    found it necessary to split off geometric realization of categories as a separate entry, recorded Quillen’s theorems A and B there

    all very briefly. I notice that David Roberts has more on his personal web (have included it as a reference)

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMay 30th 2011

    The naming conventions of nnLab I think prefer singular expressions. Like geometric realization of a category.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011

    Not sure about this. “geometric realization” is clearly singular and as an operation it is “of categories” and not of a single category.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMay 30th 2011
    • (edited May 30th 2011)

    Almost any function or operation in mathematics can be extended to and viewed as a functor on many objects. Like for Cauchy completion of a category you can say that Cauchy completion is an operation on Cat. Spectrum is an antiequivalence of categories and one still talks of a spectrum of a ring as the basic notion.

    Where do you draw the line ? What a fundamental difference you see between the geometric realization and Cauchy completion ?

    Singular makes sense.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011

    I don’t know. Plural sounds more right to me here. Just a gut feeling. For instance, in the case at hand, Quillen’s theorems apply to the geometric realization of categories not to that of a single category, where it has nothing to say.

    But I don’t mind if you feel like changing the entry title.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeMay 30th 2011

    Most what I would say about spectrum or Cauchy completion is also such. The emphasis of an entry on collective behaviour is an outcome of the page which changes through time. The naming conventions are for easier finding of a page by a user who does not know which emphasis in the treatment to find and for a new namer who will check for an expected form if it is in the nnLab already. While from the content side I can agree with you, from the organizational point not as the latter terms seem to be prevalent. The naming should not change because of presence of Quillen theorem or any other content addition in future changing emphasis I think.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011

    okay, I have at least added a redirect.

    On the general problem of users being able to find nnLab entries: this is something I feel very much concerned about. I think the best way to ensure this is to have enough cross-links in sections such as “Related concepts”. If we do it right, any user searching for “geometric realization of categories” should be able to find what he needs by either first going to category and then finding a corresponding link there, or first go to geometric realization and find a useful link there. You see, I am thinking we should try to ensure that the nnLab can be well navigated hierarchically. Because that seems to be the only robust way to ensure that entires will be found. That’s why I am sometimes a bit obnoxious with adding all these tocs and little tocs etc everywhere.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeMay 30th 2011

    Well related concepts is what you understand differently from me. You tend to write small parallel groups with a single common denominator. I prefer to write a list of all terms which are similar with respect to variosu criteria, not necessarily one theoretic line of systematization. Thisgives more width to difefrent users and perspectives in my opinion and I like to have it at the beginning of the literature section not to open new titles and sections. I was told two weeks ago by a colleague who says uses our pages very often that the most helpful are pages when they are not too long.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011

    You tend to write […] I prefer to write […]

    That’s why it’s good to have many authors on a wiki: different perspectives. I am just saying: let’s all try to ensure that all the relevant links are present on a page. That’s what should ensure that readers will find the information that is available.

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeMay 30th 2011
    • (edited May 30th 2011)

    I agree. But it is good to be aware of each other style. I was rethinking my style many times under unfluence of the others, including oustide users of nnLab. But the thing that we have the links does not mean that automatically everything can be found. If even people who know the nnLab well like me have little time, like I just learned some concept and have few minutes to create an entry, I will go directly looking for the exact title. A number of people created duplicates in past, so the confusion is sometimes real, and one should not expect the systematic approach every time. And if we do not use the hurry moments, I mean my experience is if I do not do somethig what I intend right away, later I do not do it.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011

    I will go directly looking for the exact title

    Yes, me too. But I do it via Google. If all entries have links to them from the relevant other entries, then Google will see them and find them. Another reason to include enough links back and forth! :-)

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 31st 2011

    Thanks, Urs! (#1)

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJun 1st 2011

    FWIW, I would go with the singular. The “geometric realization of a category” is a well-defined thing (given a category).

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2011

    added standard statements about homotopies induced by geometric realization of natural transformations

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2011

    added to geometric realization of categories the statement of “Thomason’s theorem”:

    For F:DF : D \to Cat a functor, let |F()|:DFCat||\vert F(-)\vert : D \stackrel{F}{\to} Cat \stackrel{\vert-\vert}{\to} Top be the postcomposition with geometric realization.

    Then we have a weak homotopy equivalence

    |F|hocolim|F()| \vert \int F \vert \simeq hocolim \vert F(-) \vert

    exhibiting the homotopy colimit in Top over |F()|\vert F (-) \vert as the geometric realization of the Grothendieck construction F\int F of FF.

    • CommentRowNumber16.
    • CommentAuthorKarol Szumiło
    • CommentTimeSep 9th 2013

    Definition 1 used to say “For CC a category, let C\nabla C be the poset of simplices in NCN C, ordered by inclusion. Its nerve is also called the barycentric subdivision of the nerve of CC.” I have removed the second sentence, the nerve of the poset in question is usually not (isomorphic to) the subdivision of NCN C.

    The nerve of a poset PP satisfies the property that the classifying maps of all non-degenerate simplices Δ[m]NP\Delta[m] \to N P are injective. On the other hand SdNC\mathrm{Sd} N C usually doesn’t (for example when CC is any nontrivial group).

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2013

    Thanks. I had fixed that point a while back at barycentric subdivision itself, but forgot that it was stated also here. Thanks for catching that.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2017
    • (edited Apr 26th 2017)

    I have touched the entry geometric realization of categories.

    Gave it something like an Idea-section. Tried to harmonize the notation: writing |N()|{\vert N(-)\vert} for the geometric realization instead of just ||{\vert -\vert}. Added some of the original references.