As I am studying basic topology, I encounter the following fact (Assume that $K,L$ are simplicial complexes, write $C(K)$/$C(L)$ the chain complex of chain groups of $K$/$L$, write $\partial$ the boundary operator):
A homomorphism of chain complex $\phi:C(K)\to C(L)$ satisfying $\partial \phi = \phi\partial$ will induce a homomorphism $\phi_*$ of homology groups.
Also, from a casual reading I know that in Cech cohomology, similar thing happens ($\partial\phi=\phi\partial$ on cochains induces homomorphisms on cohomology groups). The two concept are so similar except that they differ by the direction of arrows.
Then I wonder, will there be theorems about when we can bring facts in one notion to its "dual" notion, or when two "dual" notions will have similar facts? I am not familiar with the exact meaning of "dual", so I am just looking forward to anything interestingly similar.