On Generic Separable Objects

The notion of separable (alternatively unramified, or decidable) objects and their place in a categorical theory of space have been described by Lawvere (see [9]), drawing on notions of separable from algebra and unramified from geometry. In [10], Schanuel constructed the generic separable object in an extensive category with products as an object of the free category with finite sums on the dual of the category of finite sets and injections. We present here a generalization of the work of [10], replacing the category of finite sets and injections by a category A with a suitable factorization system. We describe the analogous construction, and identify and prove a universal property of the constructed category for both extensive categories and extensive categories with products (in the case A admits sums). In constructing the machinery for proving the required universal property, we recall briefly the boolean algebra structure of the summands of an object in an extensive category. We further present a notion of direct image for certain maps in an extensive category, to allow construction of left adjoints to the inverse image maps obtained from pullbacks.