Intuition behind alternating sum in boundary operator definition? Boundary of simplex is formal alternating sum of its faces.
It is clear for me, that with ordinary sum main principle $dda = 0$ will not hold for ordinary sum. For every n+2-simplex $a$, there will be two copies of every it's n-subsimplex, and if we use alternating sum they all will kill each other.
But I can not understand, what all this signs really mean.
Is there reasons why boundary operator must be defined like this? Can the definition be derived from simpler properties, as for Euler charachteristic?

I see it this way:

*

*When you define an $n$-simplex using an (ordered) tuple of $n+1$ points, you implicitly define an orientation on the set of n-simplices with the same vertices : two such n-simplices have the same orientation if they are related through an even permutation.

*Consider the $(n-1)$-simplices you can obtain by removing one of the previous points. You may want to inherit the orientation defined previously to define a "compatible" orientation on those $(n-1)$-simplices.

*(*) One way to do this is to state : $[v_0, ..., v_{j-1}, v_{j-1}, ..., v_n]$ is positively oriented iff $[v_j, v_0, ..., v_{j-1}, v_{j-1}, ..., v_n]$ is positively oriented — I'm using the notations of https://en.wikipedia.org/wiki/Simplex#Algebraic_topology.

*If you take that as a definition for an induced orientation, you now see that the orientation of $[v_0, ..., v_{j-1}, v_{j-1}, ..., v_n]$ is related to the one of $[v_0, ..., v_{j-1}, v_j, v_{j-1}, ..., v_n]$ by a factor $(-1)^j$ (corresponding to $j$ transpositions), which justifies the formula for the boundary.

Now a question is why (*) should be a good way to define an induced orientation ?
It actually gives a definition of an induced orientation compatible with the one for a manifold with boundary.
