Abstract: The notion of separable (alternatively unramified ,o rdecidable) objects and their place in a categorical theory ofspace 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 ofthe f ree category with finite sums on the dual ofthe category offinite sets and injections. We present here a generalization ofthe work of(10), replacing the category offinite 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 ofthe summands ofan 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.
Publication Year: 1998
Publication Date: 1998-01-01
Language: en
Type: article
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Cited By Count: 2
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