Wait until you learn about modern homotopy theory. It turns out that basically every cohomology theory can be described as homotopy classes of maps into some s

Wait until you learn about modern homotopy theory. It turns out that basically every cohomology theory can be described as homotopy classes of maps into some space. The standard cohomology groups with integer coefficients H^n(X; Z), those are just homotopy classes of maps from X into the Eilenberg-MacLane K(Z,n). Want K-theory instead? No problem. That cohomology group K(X) is homotopy classes of maps [X, Z x BU], where Z is the integers and BU is the classifying space for the unitary group. It turns out essentially all cohomology theories are represented this way. h(X) = [X,S_h], where [] is homotopy classes of maps and S_h is a topological space called the spectrum of the cohomology theory. Even wackier, the various algebra operations you can do on cohomology classes are mirrored by / derived from algebraic operations on the spectra. Which means that much of the machinery of algebraic geometry can be used to study cohomology theories.