In this lecture we switch our fascination from the functor $\square \otimes A$ to the functor $\mathrm{Hom}( \square, A)$.

This first requires us to take a more open-minded point of view towards functors, thus obtaining contravariant functors, where all the arrows get reversed.

We then investigate what happens when we apply $\mathrm{Hom}(\square ,A)$ to a chain complex. The result is an abomination with negative degrees, so we flip indices and end up with a cochain complex.

Applying this to the singular chain complex, we obtain the singular cohomology of a topological space, which we show satisfies the "usual" Eilenberg-Steenrod Axioms.

This lecture is mostly just definitions. 😀 (It's therefore much easier than the last four!)