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1. • CommentRowNumber1.
   • CommentAuthorchase cain
   • CommentTimeNov 21st 2019
   • (edited Nov 21st 2019)
   • PermaLink
   Author: chase cain
   Format: MarkdownItexI'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 $\operatorname{Cat}_{\ast,\ell}\to\operatorname{Cat}$. In [2], under the section "$n$-subobject classifiers" is the statement: > $E_{pt}\operatorname{Cat}\to\operatorname{Cat}$ is $\operatorname{Cat}_*\to\operatorname{Cat}$. Pullback of this gives the Grothendieck construction. Of course, the categories $\operatorname{Cat}_*$ and $\operatorname{Cat}_{\ast,\ell}$ are slightly different, only in the morphisms. My question is: can the category $\operatorname{Cat}_{\ast,\ell}$ be described as some variation of the pullback: $$E_{pt}\operatorname{Cat} = \lim([I,\operatorname{Cat}]\to\operatorname{Cat}\leftarrow pt) = \operatorname{Cat}_*?$$

   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 $\operatorname{Cat}_{\ast,\ell}\to\operatorname{Cat}$. In [2], under the section “$n$-subobject classifiers” is the statement:

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

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

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