In algebraic topology, there is a close relationship between stable homotopy theory and the study of (generalized) cohomology theories. Basically, all the cohomology theories become representable on the stable category of spectra and so, from the point of view of stable homotopy theory, the study of cohomology theories can be viewed as the study of their representing spectra.
More recently it has been discovered (through the work of Voevodsky and others) that there is an analogous situation in algebraic geometry. Keywords to look up: motivic homotopy theory, $\mathbb{A}^1$-homotopy theory. Basically, we can construct a homotopy theory for algebraic varieties and a suitable homotopy category which plays a role analogous to the stable category of spectra in topology. One thing this gives us is that it enables us to define new cohomology theories for algebraic varieties by describing their representing spectra ("motivic spectra"). For example, motivic cohomology, algebraic K-theory, and algebraic cobordism can be constructed in this way. This whole circle of ideas is closely related to recent work on motives and motivic cohomology. For example, Voevodsky's construction of a "derived category of mixed motives" is closely related to this work.
The following is a very easy-going introduction to the idea of motivic homotopy theory and is understandable even by an undergraduate:
- Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002 by Bjørn Dundas, Marc Levine, Paul Østvær, Oliver Röndigs and Vladimir Voevodsky
It is also worth reading Voevodsky's 1998 ICM address:
- Vladimir Voevodsky - A^1-homotopy theory (Proceedings of the 1998 ICM)
There is a lot more that could be said about this very interesting area of mathematics.
