To complete the question in the title: I'm currently working on simplicial manifolds, using the definition stated here: https://arxiv.org/pdf/2112.01417.pdf , page 4. That is, the simplicial manifolds are defined as a tower of manifolds with face and degeneracy maps between them, satisfying the simplicial identities.
But in another paper(https://arxiv.org/pdf/1012.4103.pdf page 4), the definition also says the face maps should be submersions. Although it seems quite intuitive, I get difficulties to prove it using the first definition.
So here is my question: are the both definitions equivalent, and if yes, could you give me a hint why?
My supposition would be that they are not the same and I guess the following is a counter-example:
Here, $X_0$ is simply $(0,1)$ and $X_2$ is the grey surface embedded in $(0,1)^2$. The face maps are simply the projection along the axes and you can see in green the image of the degeneracy map.
In this case, the segment in the left upper part would be the problem. Indeed, the projecting of this segment on the y-axe will not be a submersion, implying that $d_1$ would not be a submersion.