In simplices complex, it is known that boundary of boundary is zero
$$\partial_{n-1} \partial_{n} \sigma_{0,\ldots,n} = 0$$ where $\sigma_{0,\ldots,n}$ is a n-simplice consists of $n+1$ point, which by definition, it is a convex hull of those point.
Someone showed that on the youtube video about $$\sigma_1 + \sigma_1 = 0$$
But why? I cannot see it even in intuition.
That video is working with chains with coefficients in $\mathbb{Z}/2\mathbb{Z}$. So, $\sigma_1+\sigma_1=2\sigma_1$ is $0$ since the coefficients on the individual simplices are considered as integers mod $2$ and so $2$ is the same as $0$.
It is possible to instead consider chains with coefficients in $\mathbb{Z}$, and then $\sigma_1+\sigma_1$ would not be $0$. If you do that, though, you have to modify the definition of the boundary operators to include some minus signs and so the boundary of a loop based at $\sigma_1$ would turn out to be $\sigma_1-\sigma_1$ instead of $\sigma_1+\sigma_1$ (one of the endpoints gets counted with a minus sign).