Some structural similarities between uncountable sets, powersets and the universe

We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras $\mathcal{P}(\kappa )$, for a regular cardinal $\kappa$, and $\mathcal{C}(V)$ (the class of subclasses of the universe $V$), endowed with some ideals, especially the ideal $[\kappa {]}^{<\kappa }$ for $\mathcal{P}(\kappa )$, and the ideal of sets $V$ for $\mathcal{C}(V)$.