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 ordinal 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.

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.

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

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 ordinal 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.