I've been reading "MacLane's Categories for the Working Mathematician" today because I was trying to find out how the proof of the Coherence Theorem for strict monoidal categories goes, and holy crap is there a lot of stuff in that book that makes so much more sense now. Freakin' everything is a monoid object in a suitable monoidal category. Categories themselves? Monoid objects in <O-Graph, ×O, id>. Monads? Monoid objects in <CC, ∘, id>. Rings? Monoid objects in <Ab, ⊗, Z>. And if it's not a monoid object, it's the action of one. Modules are actions of rings, suitable group actions are torsors, monad actions are T-algebras, which themselves include the remainder of the usual zoo of algebraic structures like semigroups, groups, etc.