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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeOct 5th 2012
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   Author: Mike Shulman
   Format: MarkdownItexRoss Tate has pointed out a mismatch in terminology: Kleisli objects and the Grothendieck construction (of a covariant Cat-valued functor) are both asserted to be "lax colimits", but they are not the same kind of colimit (the 2-cells go in different directions). Thinking about this more, I have concluded that Kleisli objects are lax colimits and the Grothendieck construction is an *oplax* colimit. I wrote a bit about my reasoning [here](http://ncatlab.org/nlab/show/2-limit#lax). But before I go changing all references to the Grothendieck construction to say "oplax colimit", I thought I should do a sanity check --- does this make sense to everyone else?

   Ross Tate has pointed out a mismatch in terminology: Kleisli objects and the Grothendieck construction (of a covariant Cat-valued functor) are both asserted to be “lax colimits”, but they are not the same kind of colimit (the 2-cells go in different directions). Thinking about this more, I have concluded that Kleisli objects are lax colimits and the Grothendieck construction is an oplax colimit. I wrote a bit about my reasoning here. But before I go changing all references to the Grothendieck construction to say “oplax colimit”, I thought I should do a sanity check — does this make sense to everyone else?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 6th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI'd forgotten about it, but this question also came up on [MO](http://mathoverflow.net/questions/16930/terminology-lax-vs-oplax-colimits) over 2 years ago.

   I’d forgotten about it, but this question also came up on MO over 2 years ago.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 11th 2012
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   Author: Mike Shulman
   Format: MarkdownItexWell, nobody said anything, so I went ahead and changed the few references I could find.

   Well, nobody said anything, so I went ahead and changed the few references I could find.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 11th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexIt's okay with me. I was recusing myself from discussing it anyway. :-)

   It’s okay with me. I was recusing myself from discussing it anyway. :-)