""" I was asking what categories, eg monads, comonads, all these abstractions on the abstractions of mathematics, want, since that might help me understand how they see their purpose... """
I am unsure how to approach such a question. What I can respond to, wrt categories (mathematics), is the question of what is the content of a category. What follows I suspect you know full well, but maybe I will accidentally type something surprising. Categories consist of *objects* and most importantly *morphisms*. I emphasize morphisms because, unlike arbitrary functions, they are constrained to equationally preserve properties of and between the objects. For instance, morphisms in a category of groups, homomorphisms, are constrained to relate symmetry in one object to symmetry in another. In a topological category, the morphisms (homeomorphisms) are constrained to relate continuity in one topological space to continuity in another. Fields like algebraic topology arose by posing questions like, "How is this chain of relations between continuous objects like a chain of symmetry relations"? Answers to this question varied greatly and now we have homotopy, homology, cohomology, and each with its own special flavors (singular, simplicial,...) When I look at a chain of homology groups, I see, on the one hand, an analogy between (continuous spaces and symmetries) and on the other I see an encoding of spaces into algebra (a kind of accounting technique). The formal analogies are functors and the accounting is one way to see a purpose for the analogy.
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