Topological objects and sheaves

Abstract

In this paper we shall be concerned with a category ', previously introduced in [2]. We recall that the objects and maps in le (henceforth referred to simply as objects and maps) constitute a generalization of the concepts of topological space and continuous function. In particular, one finds that the category C of topological spaces and continuous functions may be canonically identified with a (proper) subcategory Xi of W. Moreover, it has been shown in [2] that every subcategory C0 of C admits a canonical extension to a subcategory Wo of W, which in some sense is maximal. It may now be asked whether classical theories, defined on a subcategory C0 of C, admit a natural extension to the corresponding object category %o. This extension problem has been considered in [2] for the case of theories arising from a sheaf-valued functor. In the present paper we will describea canonical extension of such functors in full detail, and shall establish pertinent results. Some effort has been made in [2] to indicate the original motivation which has led to the definition of object categories. It has been pointed out in this connection that numerous structures (e.g., Lie groups, homogeneous spaces, foliated or affinely connected manifolds) give rise to canonical objects. The program of extending concepts of topology and geometry to object categories is therefore suggestive of various applications. At this point it seems appropriate to review in broad terms what is meant by an object, and in what respect a general object differs fundamentally from a topological space. An object X has been conceived as a pair (X,I), where X is a topological space and I a collection of local maps f: X -* X, subject to certain axioms. Given two objects X and X', one conceives a map F: X -* X' as a collection of local maps f: X -+ X', subject again to appropriate axioms. Having defined the composition of maps, one arrives at the category W. At this stage one observes that a considerable part of the structure which enters into the definition of an object X is not preserved under '-isomorphism. On the other hand, we have adopted the viewpoint that only those properties of (X,I) which are