[Koshkin]

Well, at the very least Category Theory can help keep things organized. Let me quote Tom Leinster:

K-theory and K-homology have become indispensable tools in operator theory; there is even a bivariant functor 𝐾𝐾(−,−) from the category of C-algebras to the category of abelian groups relating the two constructions, and many deep theorems can be subsumed in the assertion that there is a category whose objects are C-algebras and whose morphism spaces are given by 𝐾𝐾(𝐴,𝐵). Cyclic homology and cohomology has also become extremely relevant to the interface between analysis and topology.