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

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 $$(\sqcup{U_{i}\cap U_{j}\cap U_{k}} \substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow} \sqcup{U_{i}\cap U_{j}}\rightrightarrows \sqcup U_k)$$?

• Check last 1/4 th of page 9 and first 1/4 th of page 10 in arxiv.org/abs/1706.07152 – Praphulla Koushik Mar 3 at 13:13
• what is the 1 morphism in Cech groupoid? – Praphulla Koushik Mar 3 at 13:21
• 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 – Dmitri Pavlov Mar 3 at 14:25
• @PraphullaKoushik Thanks for the link. From the link, it seems what I guessed about the definition of Lie 2-Groupoid is right. – Adittya Chaudhuri Mar 3 at 17:15
• @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. – 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

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.

• 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. – Adittya Chaudhuri Mar 4 at 6:52
• 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... – Praphulla Koushik Mar 4 at 6:59
• I mean to say that only. – Adittya Chaudhuri Mar 4 at 7:05
• But I have said that in the above answer :O – Praphulla Koushik Mar 4 at 7:07
• 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. (: – 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.

• 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. – Dmitri Pavlov Mar 4 at 17:48
• @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. – Adittya Chaudhuri Mar 4 at 18:11
• 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. – Dmitri Pavlov Mar 4 at 18:33
• 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). – Dmitri Pavlov Mar 5 at 0:12
• 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. – 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

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.

• 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…". – Dmitri Pavlov Mar 14 at 15:43