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?