I added a remark to 2-limit.
]]>I’m convinced, at least. Thanks!
]]>This is a very common question, so maybe we should include this explanation at 2-limit.
]]>Yes, there is. For any weight , a -colimit in is defined to be a -limit in . This is a standard terminology in enriched category theory. Since a lax -weighted limit is the same as an ordinary -weighted limit, it follows that a lax colimit in is a lax limit in .
]]>Is there a reason for defining ‘lax colimit in ’ to mean ‘lax limit in ’ instead of ‘lax limit in ’? The latter definition also takes care of the unfortunate fact that lax limits in involve oplax natural transformations in …
]]>This is good, thanks. I made one additional change, moving “Lax limits” to a subsection of “Examples”. I think it’s more appropriate there, since lax limits are really a subclass of (weighted) 2-limits.
]]>I have rearranged the sections at 2-limit a bit. Check if you agree that this is better:
made “Strictness and terminology” and “Lax limits” subsections of “Definition”.
collected other sections as subsections of a new big “Examples”-section.
]]>I have added more to the section 2-Colimits in Cat.
I have also added to (infinity,1)-colimit a new section infinity-Colimits in (infinity,1)-Cat with the general statement (that the -colimit is given by formally inverting Cartesian morphisms in the -Grothendieck construction.)
]]>Looks good, thanks; I tweaked it a little more. (-:
]]>Well, yes, we should just say “2-limit”. Except that, as you say, in all of the literature “2-limit” means strict. I’ve had another go.
]]>Hmm, that’s not exactly what I had in mind. I thought that on the nLab we had decided to use “2-limit” to mean what is traditionally called a “bilimit” and eschew that misguided terminology entirely.
]]>I think that “bilimit” would be more likely to be understood, so I’ll take your “yes” as a reason to change it.
]]>@Toby: yes. I felt the need to say “non-strict” explicitly, since in all the literature about these things, “2-limit” means strict. If you can clarify the wording, please feel free.
]]>Mike, where you write “non-strict 2-limit”, can I read “bilimit”? (and therefore just “2-limit” or even “limit”, since I know that these should be maximally weak by default).
]]>Much appreciated, Mike!
]]>I created flexible limit.
]]>This MO question shows that we have not enough about examples at 2-limit.
For a second I felt energetic and started a section 2-limit – Examples – 2-Colimits in Cat but after writing one sentence I realize that I should be doing something else. Sorry.
]]>in reaction to this nCafe discussion I have added to the entry 2-limit a subsection on (2,1)-limits.
]]>(1) Yes. People who call our 2-limits “bilimits” usually say “2-limit” for what we are calling a “strict 2-limit.” The section “Strictness and terminology,” and the page strict 2-limit, say a bit about this, but perhaps not enough.
(2) I think you are absolutely right that the equivalence should be either (a) a map in one direction with the property of being an equivalence, or (b) an adjoint equivalence. I had the first one in mind, since by the bicategorical Yoneda lemma a map from left to right is the same as a “weighted cone” with vertex being the limit object; thus a 2-limit would then be a weighted cone with some universal property. But if you give an adjoint equivalence, then you instead have a weighted cone with some universal “structure” (though of course it is still a “property” in the formal sense, since it is unique up to unique isomorphism). Does that make sense? It would be good to clarify this on the page.
]]>2-limits is one of those areas that I’ve never studied in any depth, maybe largely because various people have at times sounded off ominous warnings about how tricky or subtle they are. So please excuse some naive questions:
(1) What people call “bilimits” – is this the same as what are being called 2-limits here? When people make a distinction, in what does the distinction consist?
(2) Is there any point in replacing the pseudonatural equivalence by an adjoint equivalence? The latter is sometimes a nicer concept to work with. (I should add that I am familiar with the theorem given at adjoint equivalence, which says that an equivalence may be replced by an an adjoint equivalence.)
]]>I finally got around to putting in the general definition at 2-limit. Explaining how the specific examples follow from this definition will have to wait for another time….
]]>Just testing a bug. Nothing to see, move right along.
]]>Updated the definition of pullback on the 2-limit page. I will link to the 2-pullback page and try to make notation more consistent soon.
]]>Great, thanks. I will do some clarification on the page very soon.
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