## Functoriality of Cubic Singular Homology

We want to show that the cubic singular homology we’ve constructed is actually a functor. That is, given a smooth map we want a chain map , which then will induce a map on homology: .

The definition couldn’t be simpler. We really only need to define the image of a singular -cube in and extend by linearity. And since is a function, we can just compose it with to get a singular -cube . What’s the face of this singular -cube? Why it’s

and so we find that this map commutes with the boundary operation , making it a chain map.

We should still check functoriality. The identity map clearly gives us the identity chain map. And if and are two smooth maps, then we can check

which makes this construction a covariant functor.

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