I am reading The Uses and Abuses of the History of Topos Theory by Colin McLarty. On page 353, the following is said:
An homology theory associates groups to topological spaces so that the group structures reflect topological structure. An homology theory also associates group homomorphisms to maps- a fact topologists in the 1930s used heavily but considered secondary to the association of groups to spaces. There were many homology theories, the relation between them were not well understood, and it seemed that specific calculations in homology could be made more systematic than they were ..
I want to ask, what were the different viewpoints of homology theory in 1930? How could this same idea of assigning a group to a topological space be realized in so many forms? Finally, how did these ideas get united?