. A finitary monad A on the category of globular sets provides basic algebraic operations from which more involved `pasting' operations can be derived. To makes this rigorous, we define A-computads and construct a monad on the category of A-computads whose algebras are A-algebras; an action of the new monad encapsulates the pasting operations. When A is the monad whose algebras are n-categories, an A-computad is an n-computad in the sense of R.Street. When A is associated to a higher operad (in the sense of the author) , we obtain pasting in weak n-categories. This is intended as a first step towards proving the equivalence of the various definitions of weak n-category now in the literature. Introduction This work arose as a reflection on the foundation of higher dimensional category theory. One of the main ingredients of any proposed definition of weak n-category is the shape of diagrams (pasting scheme) we accept to be composable. In a globular approach [3] each k-cell has a source ...

Abstract not found

Abstract. A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category [Str74b] and provides a self-contained development of the necessary background material on yoneda structures.

We propose a categorical definition of locally-compact Hausdorff object which gives the right notion both, for topological spaces and for locales. Stability properties follow from easy categorical arguments. The map version of the notion leads to an investigation of restrictions of perfect maps to open subspaces. AMS Subj. Class.: 18B30, 54B30, 54D30, 54D45, 54C10. Key words: closure operator, locally compact object, open map, perfect map. 1 Introduction Both, for topological spaces and for locales, locally compact Hausdorff spaces are characterized as the spaces which are openly embeddable into compact Hausdorff spaces. While in Top this is an obvious consequence of Alexandroff's one-point-compactification, in Loc one uses results of Vermeulen [10] to establish this result. In this note we show that, taking this characterization as the defining property for (Hausdorff) local compactness, one establishes practically all standard stability properties of local compactness -- with ...

Abstract. A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category [Str74b] and provides a self-contained development of the necessary background material on yoneda structures.