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:

$E_{pt}\operatorname{Cat} = \lim([I,\operatorname{Cat}]\to\operatorname{Cat}\leftarrow pt) = \operatorname{Cat}_*?$ ]]>Stub for the Milnor construction of universal principal $G$-bundles and the disambiguation page universal bundle (we should have universal vector and associated fibre bundles, not only principal and higher principal cases; and the discussion that there is no difference in classification of locally trivial bundles in the principal versus fibre bundle case if only the effective actions on typical fibers are considered). Added a redirect James Stasheff at Jim Stasheff required at Milnor construction (before the link James Stasheff did not work though many of his papers written under that version of the name are quoted in $n$Lab!).

]]>New entry Banach bundle covering for now also more special notion of Hilbert bundle and a *different* notion of Banach algebraic bundle. Sanity check is welcome!

Urs, David Roberts and I got into discussion of locally trivial noncommutative bundles in a discussion with a wrong title (see around here), so let us better move it here. There are still some of my latest posts there which Urs and David might have not yet seen.

I decided to update a bit noncommutative principal bundle, so I will start today a bit.

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