(Co)Homology

This article was written to expose the reader to new theories of homology. Homology and Cohomology are theories that yield invariants of mathematical objects. The most well-known and the first such theory was that of topological spaces.

An example of an invariant of a topological space is its fundamental group; if two spaces have non-isomorphic fundamental groups then in fact they are not homotopic–a stronger property than failing to be homeomorphic. However, this approach has serious limits as many different spaces have the same fundamental group, for example all Rn have π1(Rn) trivial. A natural remedy is to look at higher homotopy groups and for certain spaces this can be said to completely solve the problem but here we encounter the problem that such groups are completely intractable to compute. This
is where the notion of homology may be useful. For every topological space we can find N many homology groups which sort of act as linearisations of their homotopy counterparts, so we have many invariants to check if spaces are different while remaining typically much more amenable to computation. In fact we can go further and also calculate cohomology and the cohomology ring
which gives us greater precision.

In fact, the very properties which allowed various theories going by the name “homology” to be easily computed, i.e. the Eilenberg-Steenroods (to be seen in the sequel), allowed mathematicians to show that these theories were simply different ways to construct a single notion of homology, for example: simplicial, singular, Cech or cellular homology. This remarkable fact gave credence to the idea that homology should be seen as a truly canonical invariant of spaces. The success of this
theory motivated mathematicians to apply the same ideas to a very wide range of mathematical objects, in this article we will discuss four: Groups, Groupoids, Graphs, and Categories.

Although these other homologies theories may initially appear exotic, they will share a number of nice properties which single them out as being fundamentally homological in nature.