Why the third stage of Cech nerve a Lie 2-groupoid?

Question

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.

Answer 1

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

  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:

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

Answer 2

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.

Answer 3

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

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.