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added stuff to 2-groupoid.
Fixed two typos there.
Thanks.
It seems to me that most of the stuff about strict $2$-groupoids should be at your new article strict 2-groupoid. I have moved it.
Thanks, good point.
Turns out some people here at ESI are desperate to finally understand what on earth a 2-groupoid might be. It seems the $n$Lab entries in their present state are not helping them much (maybe unsurprisingly). Maybe this motivates somebody to show off his expository skills and expand the entries a bit.
Well, I could write, at bigroupoid, a detailed definition similar to the one that I wrote at bicategory. I’m not sure that this would be very enlightening, however.
In the meantime, I’ve written a definition paralleling the first definition at bicategory. So this has everything except the coherence laws, written in terms of natural transformations between functors between hom-groupoids.
[bigcategory]??
Fixed.
Thanks!
i have added now to 2-groupoid a description of 3-coskeletal Kan complexes, mentioning how the existence of the 3-simplices gives the composition operation and the unique 4-simplices filling any 3-sphere of 3-simplices the associativity coherence law.
By Duskin’s result this is equivalent to the definition of a bigroupoid, but much simpler to state.
I have added a simple example to strict 2-groupoid, in the hope that it helps.
@Urs Re: No 5 above. Can you tell us what the sticking point for people in understanding 2-groupoid seems to be. Analysing that may help us improve the entries. Perhaps an ’intro’ entry would help with more examples discussed in some detail. (I like the intro entries that have recently been written as they are more discursive and go through things at the ’for dummies’ level… more suitable for some of us :-), i.e. yours truly!)
Can you tell us what the sticking point for people in understanding 2-groupoid seems to be.
I can give some indications.
The main problem is a general unfamiliarity with the general kind of toolset used to handle these things. There are people who used to entertain attitude that category theory is all just nonsense, and so they struggle now when one comes to a situation where one needs to be able to freely talk about functors and natural transformations, say in the definition of composition in a bigroupoid, or say in the definition of simplicial sets and simplicial homotopy groups.
Can you tell us what the sticking point for people in understanding 2-groupoid seems to be.
As a certified dummy, I can say that thinking of a group as a one-object groupoid is beautiful. It makes everything click. I love it. Plus, once you understand a group as a one-object groupoid, it gives “for free” a pretty decent understanding of groupoids more generally.
However, the same is not necessarily true for 2-groupoids. I don’t think that thinking of a 2-group as a one-object 2-groupoid helps with intuition much because few if any (certainly not me) have any kind of intuition for 2-categories. Once you understand strict 2-categories, you get 2-groupoids for free, but that first step is a barrier.
You can read it over and over, but nothing clicked for me until I started drawing tons of pictures of baby strict 2-categories. Then I started getting some sense for what horizontal composition is. That is kind of the key stumbling block for dummies I think, i.e. horizontal composition. I don’t claim to have 2-groupoids fully under control yet, but horizontal composition is a barrier for dummies. The nLab page is a bit bare on the subject. If I find a moment (it is 10pm and I’m still in the office with no indication I’ll be going home any time soon) I’d try to add some of my hard-earned wisdom.
@Eric I added something to strict 2-groupoid earlier today and would welcome your reaction to it. It looked at the example of a congruence coming from a group homomorphism. The horizontal composition is from the transitive nature of the congruence and the vertical composition from the group multiplication. Everything is explicit but general (i.e. there are two groups and a homomorphism between them as input, not two specific groups …., as the point might get lost in the example.)
@Urs I suggest that I (and anyone else who feels like helping) have a go at writing something with such an audience in mind. I have not thought out what to say but they are an important group of people and it would be good to see if we can help them … mostly for the challenge.
Tim,
yes. It seems the times are such that good exposition in nLab entries of basic concepts of higher categories would find a comparatively large number of thankful readers. Something that wasn’t true in this way a while back, I think.
Yes, a Gpd-enriched category is exactly the same as a strict (2,1)-category, so a 2-groupoid = (2,0)-category is a special sort of Gpd-enriched category. Feel free to edit the page to clarify…
’Grpd-enriched groupoid’ sounds ok to me. I have a nagging thought that is probably silly. What is the usual definition of a V-enriched groupoid? I presume it must have an inversion morphism on each hom-object $C(a,b)\to C(b,a)$, but does that imply any further equations (other than the obvious one that it is an inversion!). I don’t think so, but I thought I would mention it just in case.
It would need to be the inverse! Namely, there would be a diagram expressing: $C(a,b) \stackrel{\Delta}{\to}C(a,b)\otimes C(a,b) \stackrel{id\otimes(-)^{-1}}{\to} C(a,b) \otimes C(b,a) \to C(a,a)$ factors through $I \to C(a,a)$ (and one the other way, to $C(b,b)$), but now that I write this, one needs the diagonal, which clearly doesn’t always exist. It does for the case at hand (as $Gpd$ is cartesian), though.
@David. Aha! Good point. That may explain my feeling. A nice point would be that in many contexts of homological algebra we have approximations to the diagonal and that must give an interesting structure. Is there a formulation that avoids this, I wonder?
I deleted one of the definitions of strict 2-groupoid, since it was not equivalent to the others, and added a remark about it to the “Properties” section.
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