Topologization and Functional Analytification I: Intrinsic Morphisms of Commutative Algebras

Eventually after Dieudonné-Grothendieck, we give intrinsic definitions of étale, lisse and non-ramifié morphisms for general adic rings and general locally convex rings. And we investigate the corresponding étale-like, lisse-like and non-ramifié-like morphisms for general $\infty$-Banach, $\infty$-Borné and $\infty$-ind-Fréchet $\infty$-rings and $\infty$-functors into $\infty$-groupoid (as in the work of Bambozzi-Ben-Bassat-Kremnizer) in some intrinsic way by using the corresponding infinitesimal stacks and crystalline stacks. The two directions of generalization will intersect at Huber's book in the strongly noetherian situation.

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