An axiomatic approach to analytic $1$-affineness

The notion of $1$-affineness was originally formulated by Gaitsgory in the context of derived algebraic geometry. Motivated by applications to rigid and analytic geometry, we introduce two very general and abstract frameworks where it makes sense to ask for objects to be $1$-affine with respect to some sheaf of categories. The first framework is suited for studying the problem of $1$-affineness when the sheaf of categories arises from an operation in a six-functor formalism over $\mathscr{C}$; we apply it to the setting of analytic stacks and condensed mathematics. The second one concerns $1$-affineness in the context of quasi-coherent sheaves of categorical modules over stable module categories: it simultaneously generalizes the algebro-geometric setting of Gaitsgory and makes it possible to formulate the problem also when dealing with rigid analytic varieties and categories of nuclear modules.