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• CommentRowNumber1.
• CommentAuthorAlexHoffnung
• CommentTimeMay 9th 2010
Hi everyone,

Somehow I seem to have forgotten of the existence of the nForum for the last few months until Mike Shulman reminded me yesterday. Oops!

I have a question about the pages on 2-limits and especially 2-pullbacks. First, I made a small change to the example of pullbacks on the 2-limit page. In part (1) of the universal property I just changed a 2-cell to its inverse, because it did not seem to parse otherwise. I think this equation is what is meant by "coherent" on the 2-pullback page. If so I would like to add that equation to the 2-pullback page. I guess it might not be that hard to guess what `coherent' means as an equation in this case, but it is not too helpful when looking for a precise definition. Is there a page that defines what it means for a bunch of arrows (2-cells in this case) to be coherent? I am guessing it just means that everything commutes.

My reason for being picky and a little dumb about these pages is that I have read through a bunch of the literature and was not able to find a "working" definition of something like weak pullback --- meaning the only definition I found was by Street in the paper on fibrations in bicategories, and this seems to need a bit of unpacking. Weber has a definition in his paper on strict 2-toposes, but as he says, it is not fully weak. Finally, the nLab seemed to have to most useful definition, so I wanted to make sure it was correct. Does anyone have anything to say about how closely the nLab definition matches Street's definition? Mike has told me that the nLab page is meant to be an unpacking of Street's definition, so I will try to add Street's general definition soon. Are there other references on 2-pullbacks or 2-limits that I am missing?

Thanks,
Alex
• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMay 9th 2010

Thanks for fixing the 2-cell to its inverse; that was probably my fault. I have a sloppy habit of using the same letter to notate an isomorphism and its inverse, which is perhaps excusable in private but not in public. (-: You’re right that that is what is meant by “coherent,” and it would be good to say that. Also the page 2-limit should link to 2-pullback! There is no fully general definition of “coherent,” but usually “all the diagrams you would hope to commute do” is a good guess.

Now I realize I might have lied to you, because the primary nLab definition of a 2-pullback is not exactly an unpacking of Street’s definition, i.e. of the general definition of 2-limit. Instead it omits one structure map, which is unnecessary up to equivalence, as explained here.

• CommentRowNumber3.
• CommentAuthorAlexHoffnung
• CommentTimeMay 9th 2010
Ok, omitting that structure map makes me happy though, since it makes lengthy calculations more manageable and diagrams look nicer.
• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeMay 9th 2010

Oh, absolutely.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 9th 2010

Eventually it would be nice if on top of the case-by-case definition and discussion we could have a more systematic notion of 2-limits.

There is an established and trusted notion of adjunction in a tricategory. So we can spak about the left and right 2-Kan extension along a 2-functor $C \to D$ as being the left and right adjoint to the 2-functor of precomposition with that functor. Now for $D = *$ this gives a notion of 2-limit and 2-colimit.

It would be nice to discuss how the special 2-limits currently discussed at the page 2-limit are reproduced as special cases of this. If they are.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeMay 9th 2010

The systematic notion in question is what Alex and I are referring to as “Street’s definition.” One of us will probably put it in soon.

It would be nice to discuss how the special 2-limits currently discussed at the page 2-limit are reproduced as special cases of this. If they are.

Some of them are, but some of them are not. The point is that for 2-categories, you are getting into the world of enriched (higher) category theory, where you have to use weighted limits. This remains true when you weaken everything, so you need weighted 2-limits, whereas adjoints to diagonal functors only give you unweighted ones. 2-products and 2-pullbacks are unweighted 2-limits, but 2-limits such as comma objects, inserters, and equifiers can’t be expressed as unweighted ones.

Weighted limits can always be expressed using Kan extensions along more general functors than the functor to a point; essentially you extend along the inclusion of the domain into the cograph of the weight. But for that to be true, by “Kan extension” you have to mean “pointwise Kan extension,” which it is (to me) unclear how to define without having a prior notion of limit.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 10th 2010

The systematic notion in question is what Alex and I are referring to as “Street’s definition.” One of us will probably put it in soon.

Oh, I see, I wasn’t aware of that.

• CommentRowNumber8.
• CommentAuthorAlexHoffnung
• CommentTimeMay 10th 2010

I have another question about the definition on the 2-pullback page. The second part of the universal property ask for equality of 2-morphisms up to 2-isomorphism. I would guess that this should mean I stick that 2-isomorphism between the two 2-cells in question and hope for the identity. This doesn’t work because the 2-cell called $\alpha$ doesn’t have an inverse. Am I misinterpreting the “modulo specified 2-isomorphism” terminology?

• CommentRowNumber9.
• CommentAuthorAlexHoffnung
• CommentTimeMay 10th 2010

Ok, maybe I see what modulo this 2-isomorphism means now. Given $\phi : fp \Rightarrow gq$, $\alpha: pu\Rightarrow pt$ and $\beta:qu\Rightarrow qt$, then $f\alpha = g\beta$ modulo $\phi$ if $\phi\cdot t\circ f\cdot\alpha = g\cdot\beta\circ\h\cdot u$.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeMay 10th 2010

Yes, that’s it, although I think you meant to write $\phi$ instead of $h$. Feel free to clarify that on the page.

• CommentRowNumber11.
• CommentAuthorAlexHoffnung
• CommentTimeMay 11th 2010

Great, thanks. I will do some clarification on the page very soon.

• CommentRowNumber12.
• CommentAuthorAlexHoffnung
• CommentTimeMay 11th 2010

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.

• CommentRowNumber13.
• CommentAuthorAndrew Stacey
• CommentTimeMay 13th 2010

Just testing a bug. Nothing to see, move right along.

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeNov 11th 2010

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….

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeNov 12th 2010
• (edited Nov 12th 2010)

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.)

• CommentRowNumber16.
• CommentAuthorMike Shulman
• CommentTimeNov 12th 2010

(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.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJan 22nd 2011

in reaction to this nCafe discussion I have added to the entry 2-limit a subsection on (2,1)-limits.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeSep 7th 2011

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.

• CommentRowNumber19.
• CommentAuthorMike Shulman
• CommentTimeSep 8th 2011

I created flexible limit.

• CommentRowNumber20.
• CommentAuthorTodd_Trimble
• CommentTimeSep 8th 2011

Much appreciated, Mike!

• CommentRowNumber21.
• CommentAuthorTobyBartels
• CommentTimeSep 8th 2011

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).

• CommentRowNumber22.
• CommentAuthorMike Shulman
• CommentTimeSep 9th 2011

@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.

• CommentRowNumber23.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2011

I think that “bilimit” would be more likely to be understood, so I’ll take your “yes” as a reason to change it.

• CommentRowNumber24.
• CommentAuthorMike Shulman
• CommentTimeSep 9th 2011

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.

• CommentRowNumber25.
• CommentAuthorTobyBartels
• CommentTimeSep 9th 2011

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.

• CommentRowNumber26.
• CommentAuthorMike Shulman
• CommentTimeSep 9th 2011
• (edited Sep 9th 2011)

Looks good, thanks; I tweaked it a little more. (-:

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeSep 13th 2011

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 $(\infty,1)$-colimit is given by formally inverting Cartesian morphisms in the $(\infty,1)$-Grothendieck construction.)

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeSep 13th 2011

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.

• CommentRowNumber29.
• CommentAuthorMike Shulman
• CommentTimeSep 13th 2011

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.

• CommentRowNumber30.
• CommentAuthorZhen Lin
• CommentTimeNov 2nd 2013

Is there a reason for defining ‘lax colimit in $\mathfrak{K}$’ to mean ‘lax limit in $\mathfrak{K}^{op}$’ instead of ‘lax limit in $\mathfrak{K}^{co op}$’? The latter definition also takes care of the unfortunate fact that lax limits in $\mathfrak{K}^{op}$ involve oplax natural transformations in $\mathfrak{K}$

• CommentRowNumber31.
• CommentAuthorMike Shulman
• CommentTimeNov 2nd 2013

Yes, there is. For any weight $W$, a $W$-colimit in $K$ is defined to be a $W$-limit in $K^{op}$. This is a standard terminology in enriched category theory. Since a lax $W$-weighted limit is the same as an ordinary $W^\dagger$-weighted limit, it follows that a lax colimit in $K$ is a lax limit in $K^{op}$.

• CommentRowNumber32.
• CommentAuthorMike Shulman
• CommentTimeNov 2nd 2013

This is a very common question, so maybe we should include this explanation at 2-limit.

• CommentRowNumber33.
• CommentAuthorZhen Lin
• CommentTimeNov 2nd 2013

I’m convinced, at least. Thanks!

• CommentRowNumber34.
• CommentAuthorMike Shulman
• CommentTimeNov 2nd 2013

I added a remark to 2-limit.