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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.
Oh, absolutely.
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.
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.
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.
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?
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 $\phi\cdot t\circ f\cdot\alpha = g\cdot\beta\circ\h\cdot u$.
Does this seem about right?
Yes, that’s it, although I think you meant to write $\phi$ instead of $h$. Feel free to clarify that on the page.
Great, thanks. I will do some clarification on the page very soon.
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.
Just testing a bug. Nothing to see, move right along.
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….
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.)
(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.
in reaction to this nCafe discussion I have added to the entry 2-limit a subsection on (2,1)-limits.
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.
I created flexible limit.
Much appreciated, Mike!
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).
@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.
I think that “bilimit” would be more likely to be understood, so I’ll take your “yes” as a reason to change it.
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.
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.
Looks good, thanks; I tweaked it a little more. (-:
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.)
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.
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.
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 $\mathfrak{K}$…
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 $K^{op}$.
This is a very common question, so maybe we should include this explanation at 2-limit.
I’m convinced, at least. Thanks!
I added a remark to 2-limit.
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