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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:

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:

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