Given a groupoid $G,$ one can consider the canonical epimorphism $$G\_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram formed by iterative (2-categorical) fibered products of this morphism against itself. Direct inspection shows that vertices of the $2$-cartesian cube arising from these fibered products can be identified with the (objects of) the nerve of $G$: $G\_0$, $G\_1$, and $G\_2,$ whereas the edges of the cube can be identified with the face maps of the nerve. 


My first question is: 


Why is this truncated semi-simplicial nerve popping up here? And is there anyway to see what is going on geometrically? It seems like this has to do with relating the geometry of the corner of a cube to that of a 2-simplex.


Secondly, if I am given a (weak) semi-simplicial (truncated) groupoid, that is, groupoids $H\_2$, $H\_1$, and $H\_0$ together with face maps respecting the simplicial identities up to natural isomorphism, let $H$ denote the (weak) colimit of this diagram. Let $$p:H\_\cdot \to \Delta\_{H}$$ be a colimiting cocone. What is the relationship between the (semi-simplicial) Cech nerve of $$p\_0:H\_0 \to H$$ and the original diagram?



Cartesian cubes and groupoids
The answer to your second question is that the nerve of $p\_0$ admits a map from the original diagram, which is "universal among maps into [(2,1)-congruences](http://nlab.mathforge.org/nlab/show/%28n%2Cr%29-congruence)". This is a categorified version of what happens when you take the coequalizer of a parallel pair of morphisms, then the kernel pair of that coequalizer. If the diagram you started with was already a (2,1)-congruence, then it will be equivalent to the nerve of its colimit, because Gpd is a [(2,1)-exact (2,1)-category](http://nlab.mathforge.org/nlab/show/exact+2-category).


