Biseparating maps between smooth vector-valued functions on Banach manifolds

An I-category consists all Banach manifolds as objects and subclasses of continuous functions (with some kind of smoothness) as morphisms. This notion covers, for example, the categories C^∞, C_n, C, and Lip_loc of all smooth functions, C_n -functions, continuous functions, and local Lipschitz functions. It is shown by Garrido, Jaramillo and Prieto in 2000 that two C^∞ -smooth Banach manifolds X and Y are C^∞ -diffeomorphic to each other if and only if there is an algebra isomorphism from C^∞(X,R) onto C^∞(Y,R). We extend this result to general abstract I-categories, and from algebra isomorphisms of scalar functions to the maps which are linear, bijective and separating, between vector-valued functions.