I am confused with the adjoint of the coboundary map as described in this article (https://magnus.ece.gatech.edu/Papers/MuhammadEgerstedtMTNS06.pdf, page 4).
The adjoint of the coboundary map is defined as $$\delta_k^*=\partial_{k+1}.$$ This is all well and good.
What confuses me is the domain and codomain of $\Delta_k:C^k(X,\mathbb{R})\to C^k(X,\mathbb{R})$, where $$\Delta_k=\delta_{k-1}\delta^*_{k-1}+\delta_k^*\delta_k.$$
The domain and codomain of the boundary/coboundary maps are:
$$\partial_k:C_k(X)\to C_{k-1}(X)$$ and $$\delta_i:C^i(X;\mathbb{R})\to C^{i+1}(X;\mathbb{R}).$$
This doesn't seem to be compatible with the domain and codomain of $\Delta_k$, does it?
My guess is that $\delta_k^*=\partial_{k+1}$ refers to the formula, not the domain/codomain. So for instance $[v_0,v_1]^*$ is the cochain that is 1 on the edge $[v_0,v_1]$ and 0 otherwise, then $$\partial_0^*[v_0,v_1]^*=\partial_1[v_0,v_1]^*=v_1^*-v_0^*$$
Is that the correct interpretation?
Thanks for any enlightenment.