What does $C_{-1}(X)$ mean?

Question

I recently came across the formula that the Euler characteristic is equal to $$\sum\limits_{i=-N}^N (-1)^i\dim C_i(X)$$

For this to make sense, $C_{-1}(X)$ would have to exist. What would that be? What is a $-1$-simplex? A $1$-simplex with an odd permutation of vertices?

Answer 1

If $X$ is a topological space, it is standard to put $C_i(X)=0$ whenever $i<0$.