In this lecture we show that $H_n$ induces a functor $\mathsf{hTop} \to \mathsf{Ab}$ by proving the "homotopy axiom". The reason for the word "axiom" will become clear at the end of the course.

Indeed, there are many "different" (co)homology theories that you may have heard mentioned in some of your other courses:

• singular homology (the one we just did),
• simplicial homology,
• cellular homology (we'll do this in Lecture 20),
• de Rham cohomology (normally studied in Differential Geometry),
• Morse homology,
• Floer homology (my favourite),
• Čech cohomology,
• Borel-Moore homology,
• blahblah homology,
• ...

Luckily they all turn out to be the same (at least for nice topological spaces). This is because one can characterise "a homology theory" axiomatically, and then prove that any two such theories are isomorphic. We will do so in Lecture 22.