In the page https://ncatlab.org/nlab/show/Lie+2-groupoid the Lie 2-groupoid is defined as the 2 truncated $\infty$-Lie groupoid.

I am not much comfortable with the language of higher category theory yet. But after reading some links in ncatlab I felt Lie 2-Groupoid is the same as a 2-groupoid( a 2 category whose both 1 morphisms and 2 morphisms are invertible) internal to the category of smooth manifolds.

Am I right??

Now on the page https://ncatlab.org/nlab/show/infinity-Chern-Weil+theory+introduction#Cech2Cocycles it is mentioned that the Cech groupoid of a manifold $X$ with a cover $U_{\alpha}$ can be thought as a Lie 2 -groupoid by considering the third stage of the full Cech Nerve.

I can understand that the Cech Groupoid $(\sqcup{U_{i}\cap U_{j}}\rightrightarrows \sqcup U_k)$ is a Lie groupoid. But I am not able to understand how $(\sqcup{U_{i}\cap U_{j}\cap U_{k}} \substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow} \sqcup{U_{i}\cap U_{j}}\rightrightarrows \sqcup U_k)$ is a Lie 2-groupoid (in the sense I have understod the definition of Lie 2-groupoid). Infact I am not able to guess what can be the 2-morphisms in $(\sqcup{U_{i}\cap U_{j}\cap U_{k}} \substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow} \sqcup{U_{i}\cap U_{j}}\rightrightarrows \sqcup U_k)$ (when described as a Lie 2-groupoid).

So is my understanding of Lie 2-groupoid wrong? If not then what is the 2-categorical description of $(\sqcup{U_{i}\cap U_{j}\cap U_{k}} \substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow} \sqcup{U_{i}\cap U_{j}}\rightrightarrows \sqcup U_k)$?

Thanks in Advance.

  • 2
    $\begingroup$ Check last 1/4 th of page 9 and first 1/4 th of page 10 in arxiv.org/abs/1706.07152 $\endgroup$ – Praphulla Koushik Mar 3 at 13:13
  • $\begingroup$ what is the 1 morphism in Cech groupoid? $\endgroup$ – Praphulla Koushik Mar 3 at 13:21
  • 2
    $\begingroup$ The space of all 2-morphisms in the Lie 2-groupoid under consideration is the disjoint union of U_i ∩ U_j ∩ U_k. The correspondence between various flavors of 2-categories and simplicial objects is explained in ncatlab.org/nlab/show/geometric+nerve+of+a+bicategory $\endgroup$ – Dmitri Pavlov Mar 3 at 14:25
  • $\begingroup$ @PraphullaKoushik Thanks for the link. From the link, it seems what I guessed about the definition of Lie 2-Groupoid is right. $\endgroup$ – Adittya Chaudhuri Mar 3 at 17:15
  • 1
    $\begingroup$ @AdittyaChaudhuri: No, it doesn't work this way. The 2-categorical model used here is simplicial, so a 2-morphism is an arrow fh→g, where f, g, h are 1-morphisms given by the three simplicial faces of this 2-simplex. In the case under consideration, these three faces are given by mapping U_i∩U_j∩U_k into U_j∩U_k, U_i∩U_k, and U_i∩U_j respectively. $\endgroup$ – Dmitri Pavlov Mar 3 at 21:06

This is not an answer, just too long for a comment (could be slightly misleading). This is precisely what Dimitri Pavlov has mentioned in his comment.

In general, the description of $2$-category $\mathcal{C}$ goes as follows.

  • a collection of objects.. Let $A$ be an object of $\mathcal{C}$...

  • a $1$-morphism is between "two" objects $A\rightarrow B$...

  • a $2$-morphism is between "two" $1$-morphisms as in the following diagram enter image description here

    In simplicial model, $1$-morphism is between two objects. But, $2$-morphism is not betwen two $1$-morphisms but between three $1$-morphisms as in the following diagram: enter image description here

By definition, Cech nerve is a simplicial object. So, $2$-morphism is between three $1$-morphisms.

| cite | improve this answer | |
  • $\begingroup$ Yes, I can understand! But I am trying to now understand how we are seeing the 3rd stage of simplical object Cech Nerve as a 2 categorical model. $\endgroup$ – Adittya Chaudhuri Mar 4 at 6:52
  • $\begingroup$ I don't know if I understand your question correctly.. You do not see third stage as a 2 category.. you see first second third stage combined together as a 2 category... $\endgroup$ – Praphulla Koushik Mar 4 at 6:59
  • $\begingroup$ I mean to say that only. $\endgroup$ – Adittya Chaudhuri Mar 4 at 7:05
  • $\begingroup$ But I have said that in the above answer :O $\endgroup$ – Praphulla Koushik Mar 4 at 7:07
  • $\begingroup$ yes that is the same thing Dimitri Pavlov mentioned before in the comment. But I am trying to understand how it is following from the geometric nerve construction of bicategories. I almost got it. (: $\endgroup$ – Adittya Chaudhuri Mar 4 at 7:16

Since its too long for a comment, I had to post it as an answer in response to the comment made by Sir Dmitri Pavlov.

I got what you mentioned. You mean to say that we can treat a $(\sqcup{U_{i}\cap U_{j}\cap U_{k}} \substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow} \sqcup{U_{i}\cap U_{j}}\rightrightarrows \sqcup U_k)$ as a part of simplicial set data or in other words the whole Cech nerve can be described as a simplicial set in the following way:

$[o]\mapsto \sqcup{U_{i}}$,

$[1] \mapsto \sqcup{U_{i}\cap U_{j}}$

$[2]\mapsto \sqcup{U_{i}\cap U_{j}\cap U_{k}}$

and so on... (where $[0],[1],[2],..$ are objects of the simplex category).

where we treat elements of $\sqcup{U_{i}}$ as our objects, elements of $\sqcup{U_{i}\cap U_{j}}$ as our 1-morphisms(two face maps $d_{0}$ and $d_{1}$ are source and targets), elements of $ \sqcup{U_{i}\cap U_{j}\cap U_{k}}$ as 2 morphisms (here we consider 3 face maps $d_{0}$,$d_{1}$, and $d_{2}$).

Now I have the following doubt:

We know that there exist full, faithful Nerve functors $N_1:cat\rightarrow Ssets$, $N_{2}:Bicat\rightarrow Ssets$,.... and so on....(I am sure about $N_{1}$ and $N_{2}$ but not about higher values).

Does there exist a $c\in Bicat$ such that $N_{2}(c)$ is isomorphic to the Cech Nerve? Then we can think that $c$ as my Lie 2-Groupoid (with appropriate internalization) which represents the Cech Nerve.

| cite | improve this answer | |
  • 1
    $\begingroup$ Nerves of bicategories were characterized by Duskin in Theorem 8.6 of his paper Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. However, there is no reason to prioritize the bicategorical model over the simplicial model; in fact, the latter is what typically is used in the realm of higher Lie theory and stacks. $\endgroup$ – Dmitri Pavlov Mar 4 at 17:48
  • $\begingroup$ @DmitriPavlov Thanks. So that means in the construction of Principal 2-bundle as mentioned in ncatlab.org/nlab/show/… the morphism $g:C(U)\rightarrow B^{2}U(1)$ is actually a morphism of simplicial sets?(where C(U) is the Cech Nerve and $B^{2}U(1)$ is the one object 2 groupoid)...Then I guess they actually treated $B^{2}U(1)$ as the nerve of Lie 2-groupoid $B^{2}U(1)$?? But then it means they actually consider the whole Cech nerve ( not only upto its 3rd stage) as the former is a simplicial set and the later is not. $\endgroup$ – Adittya Chaudhuri Mar 4 at 18:11
  • 1
    $\begingroup$ Not simplicial sets, but rather simplicial presheaves on the site of smooth manifolds. Both Č(U) and B^2 U(1) are such simplicial presheaves. There is no need to consider the whole Čech nerve because the target B^2 U(1) is 2-truncated, so only simplicial levels 0, 1, 2, and 3 of Č(U) matter. $\endgroup$ – Dmitri Pavlov Mar 4 at 18:33
  • 1
    $\begingroup$ All simplicial levels of the Čech nerve are nontrivial. However, if you map into a 2-truncated object, then all simplicial levels of the source above level 3 get thrown away, whereas the 3rd level gets collapsed onto the 2nd (i.e., two 2-simplices get identified if they are connected by a homotopy presented by a 3-simplex). $\endgroup$ – Dmitri Pavlov Mar 5 at 0:12
  • 1
    $\begingroup$ A high-level discussion can be found in ncatlab.org/nlab/show/…. The article by Rezk math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf contains details in Section 7. $\endgroup$ – Dmitri Pavlov Mar 5 at 14:21

First, let me elaborate on how we obtain a groupoid by the 2-truncation of Čech nerve. Then in a similar manner, you can look at 3-truncation, 4-truncation and so on which in return produces 2-groupoid and its higher-order counterparts. In fact, the full Čech nerve is an example of an infinity groupoid (a simplicial set).

Given a manifold $M$ and a good open cover $\{U_i\}$ on it, the $2$-coskeleton $\text{Č}\left(\{U_i\}\right)_{\le2}$ of the full Čech nerve of the covering form a groupoid usually called the Čech groupoid or covering groupoid.

Its set of objects is the disjoint union $\bigsqcup_{i}U_i$ of the covering patches (open sets), and the set of morphisms is the disjoint union of the intersections $\bigsqcup_{i,j}U_i\cap U_j$ of these patches, i.e., an object is a pair $(x,i)=x_i$ where $x\in U_i$ and there is a unique morphism $(x,i,j)=x_{ij}:x_j\to x_i$ for all pairs of objects with $x\in U_i\cap U_j=U_i\underset{M}{\times}U_j.$ The composition (multiplication) of morphisms is a commutative triangle

enter image description here

satisfying $$x_{ij}x_{jk}=x_{ik}$$ for all $x\in U_i\cap U_j\cap U_k.$ In other words $m(x_{jk},x_{ij})=x_{ik}$ The source and target maps are given by $$s(x_{ij})=x_j,\, t(x_{ij})=x_i$$ for all $x\in U_i\cap U_j.$ Finally, in order to be a Lie groupoid you need to show that these structure maps are contunuous and it is enough to verify this for the source map.

| cite | improve this answer | |
  • 1
    $\begingroup$ How does this answer the question? The OP already knows how to interpret the 2-coskeleton as a Lie groupoid, the question was how to generalize this to the 3-coskeleton. This is precisely what you omitted when you said "Then in a similar manner, you can look at…". $\endgroup$ – Dmitri Pavlov Mar 14 at 15:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.