Universal Property of Non-archimedean Analytification

1.1. Motivation. Over C and over non-archimedean fields, analytification of algebraic spaces is defined as the solution to a quotient problem. Such analytification is interesting, since in the proper case it beautifully explains the essentially algebraic nature of proper analytic spaces with “many” algebraically independent meromorphic functions. (See [A] for the complex-analytic case, and [C3] for the non-archimedean case.) Working with quotients amounts to representing a covariant functor. Our aim is to characterize analytification of algebraic spaces via representing a contravariant functor, generalizing what is done for schemes. In the remainder of §1.1, we review the situation in the case of schemes, and then address the difficulties which arise for algebraic spaces (especially over non-archimedean fields). In particular, we will explain why a certain naive approach to the non-archimedean case (using functors on affinoid algebras) is ultimately not satisfactory. Our main theorem is stated in §1.2. For a scheme X locally of finite type over C, the analytification X can be defined in two ways. In the concrete method, we choose an open affine cover {Ui} and use a closed immersion of each Ui into an affine space to define U i as a zero locus of polynomials in a complex Euclidean space. These are glued together, and the result is independent of {Ui}. A more elegant approach, pushing open affines into the background and functoriality into the foreground, is to use a map iX : X → X that exhibits X as the solution to a universal mapping problem: it is final among all morphisms Z → X where Z is a complex-analytic space and morphisms are taken in the category of locally ringed spaces of C-algebras. In other words, (X, iX) represents the contravariant functor Hom(·, X) on the category of complex-analytic spaces.