I am working on the following exercise: Show that a simplicial map induces a map on chain complexes (hence maps on homology between simplicial complexes).
Here is what I have so far...
Let $K, L$ be abstract simplicial complexes, $\phi: K \rightarrow L$ a simplicial map between them. Then for each $i \in \mathbb{N}$, $\phi$ induces a chain map $\phi_i : C_i (K) \rightarrow C_i (L)$ given by $ \phi_i (\langle v_0, \ldots, v_i \rangle) = \left\{\begin{array}{ll}\langle \phi(v_0), \ldots, \phi(v_i) \rangle, & \mbox{ if the } \phi(v_j) \mbox{ are distinct} \\[30pt] 0, &\mbox{otherwise}.\end{array}\right. $ To see that $\phi_i$ is a chain map, it suffices to check that $\phi_*$ commutes with the boundary maps $\partial_1 : C_i (K) \rightarrow C_{i-1} (K) $ and $\partial_2 : C_i (L) \rightarrow C_{i-1} (L) $, i.e., $\partial_2 \circ \phi_i = \phi_i \circ \partial_1 : C_i (K) \rightarrow C_{i-1} (L).$ Begin with the LHS first: $ \partial_2 \circ \phi_i (\langle v_0, \ldots, v_i \rangle) = \left\{\begin{array}{ll} \sum_{j} (-1)^j \langle \phi(v_0), \ldots, \widehat{\phi(v_j)}, \ldots, \phi(v_i) \rangle, & \mbox{ if the } \phi(v_j) \mbox{ are distinct} \\[30pt]0, &\mbox{otherwise}.\end{array}\right. $ I am STUCK on making sense of casework for the RHS, and I am posting this question to see if anyone visiting can help me in this endeavor. Any help would be greatly appreciated.