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    • CommentRowNumber1.
    • CommentAuthorchase cain
    • CommentTimeNov 21st 2019
    • (edited Nov 21st 2019)

    I’m currently wrestling with the ideas in the following two pages:

    1. Grothendieck+construction
    2. generalized+universal+bundles

    Here are the specifics: In [1] the Grothendieck construction is described as the (strict) pullback of the universal Cat-bundle Cat *,Cat\operatorname{Cat}_{\ast,\ell}\to\operatorname{Cat}. In [2], under the section “nn-subobject classifiers” is the statement:

    E ptCatCatE_{pt}\operatorname{Cat}\to\operatorname{Cat} is Cat *Cat\operatorname{Cat}_*\to\operatorname{Cat}. Pullback of this gives the Grothendieck construction.

    Of course, the categories Cat *\operatorname{Cat}_* and Cat *,\operatorname{Cat}_{\ast,\ell} are slightly different, only in the morphisms. My question is: can the category Cat *,\operatorname{Cat}_{\ast,\ell} be described as some variation of the pullback:

    E ptCat=lim([I,Cat]Catpt)=Cat *?E_{pt}\operatorname{Cat} = \lim([I,\operatorname{Cat}]\to\operatorname{Cat}\leftarrow pt) = \operatorname{Cat}_*?