# A question on the Grothendieck construction

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.

• 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). – Michal R. Przybylek Apr 3 '14 at 0:53

• 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 $A^{op}\times A\to Cat$. – Ma Ming Apr 3 '14 at 10:34