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1. • CommentRowNumber1.
   • CommentAuthorFinnLawler
   • CommentTimeDec 20th 2011
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   Author: FinnLawler
   Format: MarkdownItexEverybody knows that if C is a category and $D \colon C \to Cat$ is a pseudofunctor then the lax colimit of D is its category of elements $\int D$ and its pseudo colimit is the category $\int D[S^{-1}]$ got by formally inverting the class S of opcartesian morphisms. Now suppose C is a bicategory and D is as before. I suspect that the (fully weak) colimit of D might be got by * taking $\int D$, to get a bicategory; * applying $\pi_0$ homwise to get a category; and finally * inverting the (images of the) opcartesian 1-cells in $\int D$. Before I work out the details, does anyone know if this has been written down before, or if I'm barking up the wrong tree entirely?

   Everybody knows that if C is a category and $D \colon C \to Cat$ is a pseudofunctor then the lax colimit of D is its category of elements $\int D$ and its pseudo colimit is the category $\int D[S^{-1}]$ got by formally inverting the class S of opcartesian morphisms.

   Now suppose C is a bicategory and D is as before. I suspect that the (fully weak) colimit of D might be got by

   • taking $\int D$, to get a bicategory;
   • applying $\pi_0$ homwise to get a category; and finally
   • inverting the (images of the) opcartesian 1-cells in $\int D$.

   Before I work out the details, does anyone know if this has been written down before, or if I’m barking up the wrong tree entirely?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 20th 2011
   • (edited Dec 20th 2011)
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   Author: Urs
   Format: MarkdownItex> Everybody knows By the way, we have only brief remarks about this in the relevant $n$Lab entries [here](http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit) and [here](http://ncatlab.org/nlab/show/2-limit#2ColimitsInCat). Maybe somebody feels like adding in more details? I am sure there are many readers out there who would appreciate it (those who are not "everybody" ;-)

   Everybody knows

   By the way, we have only brief remarks about this in the relevant $n$Lab entries here and here.

   Maybe somebody feels like adding in more details? I am sure there are many readers out there who would appreciate it (those who are not “everybody” ;-)

3. • CommentRowNumber3.
   • CommentAuthorFinnLawler
   • CommentTimeDec 20th 2011
   • PermaLink
   Author: FinnLawler
   Format: MarkdownItexOK, consider your hint dropped :) I've lots to do at the moment, but I'll put the stuff I'm working on onto my personal web at least, and I'll try to add background material to the main Lab.

   OK, consider your hint dropped :)

   I’ve lots to do at the moment, but I’ll put the stuff I’m working on onto my personal web at least, and I’ll try to add background material to the main Lab.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeDec 20th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThere's a generalization of the classical fact to $(\infty,1)$-categories in section 3.3.4 of [[Higher Topos Theory]]. If you're willing to identify locally groupoidal bicategories with certain $(\infty,1)$-categories, then I think that implies that colimits in $Cat$ of diagrams on such $C$ can be computed by making the Grothendieck construction, inverting opcartesian arrows, then (since $Cat$ is reflective in $(\infty,1)Cat$) applying $\pi_0$ homwise to get a category (reversing the order of your last two steps). But the version for non-locally-groupoidal $C$, I don't remember seeing anywhere.

   There’s a generalization of the classical fact to $(\infty,1)$-categories in section 3.3.4 of Higher Topos Theory. If you’re willing to identify locally groupoidal bicategories with certain $(\infty,1)$-categories, then I think that implies that colimits in $Cat$ of diagrams on such $C$ can be computed by making the Grothendieck construction, inverting opcartesian arrows, then (since $Cat$ is reflective in $(\infty,1)Cat$) applying $\pi_0$ homwise to get a category (reversing the order of your last two steps). But the version for non-locally-groupoidal $C$, I don’t remember seeing anywhere.

5. • CommentRowNumber5.
   • CommentAuthorFinnLawler
   • CommentTimeDec 23rd 2011
   • PermaLink
   Author: FinnLawler
   Format: MarkdownItexOK, I think this works, so I've added it to [[2-limit]], in the section [2-colimits in Cat](http://ncatlab.org/nlab/show/2-limit#2ColimitsInCat). But it's very late at night here, so a second opinion would be nice, if anyone has the time.

   OK, I think this works, so I’ve added it to 2-limit, in the section 2-colimits in Cat. But it’s very late at night here, so a second opinion would be nice, if anyone has the time.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeDec 23rd 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think I believe it. Very nice!

   I think I believe it. Very nice!