I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow:

The argument works by showing that there is a chain homotopy between the map $\partial_0^*$ and $\partial_1^*$ induced by the $0$ and $1$-section:

I wonder if this proof work for the case $I$ is a general interval object for a site. I don't know how to define such a "vertex" for a general cosimplicial object. What is the proof for $Sing^IX$ is $I$-invariant for an interval object?

I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow:

The argument works by showing that there is a chain homotopy between the map $\partial_0^*$ and $\partial_1^*$ induced by the $0$ and $1$-section:

I wonder if this proof works for the case $I$ is an interval object for a site, as defined in Section 2.3, Morel and Voevodsky. If it does, how to define vertex and simplicial decomposition for a general cosimplicial object. I suppose that to define a vertex, one needs the presheaf (or sheaf) $I$ to be a presheaf of group. If not, What is the proof for $Sing^IX$ is $I$-invariant for an interval object?