(Co)Homology: From topology to the rest of mathematics?

Question

I can appreciate how (co)homology arose in the context of topology/geometry. Trying to get a handle on the handles of spaces leads one to this idea. It's not obvious, but I can see how this would lead you to homology.

But how does one motivate/understand the expansion into other areas of mathematics? From the geometric/topological point of view simplices seem to be of fundamental importance. Yet, in all other homological theories (groups, sheaves etc.) I don't see simplices. Groups are not broken down into simplices for instance. All I see are these chains of morphisms; how did they take center stage?

Could someone please comment on how it might seem natural to arrive at these different homologies? Or point me in the direction of a reference which discusses how group homology arose from the more geometric simplicial/singular homology theories?

Thanks

Answer 1

I'd recommend Weibel’s History of homological algebra (1999)(pdf). He describes many threads, such as roots of group cohomology in Hurewicz’s observation that cohomology of an aspherical space $Y$ depends only on (what we now call cohomology of) its fundamental group $\pi=\pi_1(Y)$:

[Since] homology and cohomology groups of $Y$ (with coefficients in $A$) were independent of the choice of $Y$, Eilenberg and Mac Lane took them as the definition of $H_n(\pi; A)$ and $H^n(\pi; A)$. To perform computations, Eilenberg and Mac Lane chose a specific abstract simplicial complex $K(\pi)$ for the aspherical space $Y$. (...) Thus one way to calculate the cohomology groups of $\pi$ was to use the cellular cochain complex of $K(\pi)$, whose $n$-chains are functions $f : \pi^q \to A$.

So the simplices are not to be found in $\pi$ but in $K(\pi)$. Another crucial thread is well summarized in the first lines of Chevalley-Eilenberg (1948):

[We] give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions concerning Lie algebras. This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham’s theorems (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the consideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group.

Answer 2

As a complement to the answer provided by Francois Ziegler, I would add the first three paragraphs of Homological Algebra (1956), by Henri Cartan and Samuel Eilenberg:

During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to present a unified account of these developments and to lay the foundations of a full-fledged theory.

The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments. We present herein a single cohomology (and also a homology) theory which embodies all three; each is obtained from it by a suitable specialization.

This unification possesses all the usual advantages. One proof replaces three. In addition an interplay takes place among the three specializations; each enriches the other two.