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The usual Grothendieck construction is for pseudofunctors $I^{op}\to Cat$, where $I$ is a 1-category and $Cat$ is the 2-category of 1-categories. The Grothendieck construction produces a category with a functor into $I$, and the essential image is the Grothendieck fibrations over $I$.

My question: if we allow $I$ to be a bicategory, consider still $I^{op}\to Cat$. Is there a Grothendieck construction that produces a bicategory (?) with a functor into $I$? And what is the essential image of this construction (if there is one)? A quick candidate is the lax colimit of this functor, but I do not known if it is useful.

Added. We could go one step lower. Consider functor $I^{op}\to Set$, where $I$ is a 1-category. This is the same as an action of $I$. If I am not wrong, then the Grothendieck construction for $ I^{op}\to Set\to Cat$ will produces the action (translation) category.

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    $\begingroup$ Your quick candidate is almost the right one --- since you take a colimit in Cat, you cannot obtain a non-trivial 2-category. The right way is to postcompose $I^{op}\to Cat$ with the embedding $Cat \to 2Cat$ and take the lax colimit. This construction is sometimes called a "2-Grothendieck construction" (notice, that in case of 2-categories, there are four different Grothendieck constructions). $\endgroup$
    – Michal R. Przybylek
    Commented Apr 3, 2014 at 0:53

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The bicategory of elements of a Cat-valued functor is defined in e.g. Street's Fibrations in bicategories; it's the same as the usual one, with 2-cells as described here. Its property of classifying lax transformations is also mentioned on that page (this page describes that property in the usual setting; it is not hard to generalize it).

Baković's preprint Fibrations of bicategories (PDF here) goes into lots more detail on the Grothendieck construction for bicategory-valued functors. I don't know if it characterizes the fibrations that arise from category-valued functors (a quick text search suggests not), but one plausible guess is that they should be the ones with locally discrete fibres.

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    $\begingroup$ Thanks a lot. By the way, I have a vague feeling that the 2-coend decried here mathoverflow.net/questions/106358/… might be the (truncated) diagonal part of the Grothendieck construction for $A^{op}\times A\to Cat$. $\endgroup$
    – Ma Ming
    Commented Apr 3, 2014 at 10:34
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    $\begingroup$ My intuition is that your intuition is right - depending on the grothendieck construction used you'd end up with lax/colax variants of a 2-coend. I'll have to think about this a bit: There's actually two grothendieck constructions for such distributors: mathoverflow.net/questions/128393/… $\endgroup$
    – Gerrit Begher
    Commented Apr 4, 2014 at 15:59

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