Simplicial Homology Boundary Map applied to Single Vertex

Question

For the boundary map of a 1-simplex $[v_0,v_1]$, we have that $$\partial_1[v_0,v_1]=[v_1]-[v_0].$$

I am curious what exactly is $\partial_0 [v_0]$, is it a empty set $\emptyset$, or empty simplex $[]$, or just zero?

Thanks.

Answer 1

The image of an n-simplex under the boundary map is an (n-1)-simplex. A vertex is a 0-simplex. So it's image should be a (-1)-simplex, which is in some conventions defined as an empty set.