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A map $\phi_*:C_*\rightarrow D_*$ of chain complexes is a chain map if it intertwines the boundary operator, i.e. $\phi\partial=\partial \phi$. It is well known that such a map descents to homology, because it maps boundaries to boundaries $\phi(\partial \omega)=\partial (\phi\omega)$, and cycles to cycles, i.e. if $\partial \omega=0$, then $\partial\phi\omega=0$.

Today I encountered a map with a similar property, but it anti-intertwined the boundary, $\phi\partial=-\partial \phi$. One readily verifies that this map also descents to homology. From such a map, note can construct a chain map by setting $\psi_k=(-1)^k\phi_k$.

The category of chain complexes consists of chain complexes, along with chain maps. My question is: why don't we allow the latter type of maps as well? Is there a good reason for this, or don't we do this, because we can always construct a chain map from it? This seems to me a bit strange, because the choice is non-canonical. Another choice, namely $\psi_k=(-1)^{k+1}\phi_k$ works just as well.

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Isn't possible to see $\phi_\ast$ even as a chain map from $(C,\partial)$ to $(D,-\partial)?$ –  Giuseppe Tortorella Aug 9 '12 at 11:08
@Giuseppe, sure but those are different chain complexes. I was just wondering if there was a fundamental reason not to consider these maps. –  Thomas Rot Aug 9 '12 at 20:39
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