In a post below, I mentioned algebraic topology in computer science. A nice application of alg-top is for study of concurrency in distributed systems. For instance, one approach is to consider execution traces of a computational system being represented by time-directed paths through a space, and then to use alg-top methods to ask and answer questions about the structure of this space.
This approach leads naturally to a concept of homotopies of paths, equivalence classes of paths which may be transformed into one another via other paths in the space. What is different from traditional homotopy theory is that the paths are directed, and so these are referred to as directed homotopies or dihomotopies. Paths which are not dihomotopically equivalent represent execution traces on which there are events not reachable from one to another. For more on this, start with the GETCO conference pages.