In order to solve domain equations over 1-bounded compact metric spaces, xed points of functors on categories of 1-bounded compact metric spaces are studied. Two categories of 1-bounded compact metric spaces are considered: KMS and KMS E. In both categories, objects are isomorphic if and only if they are isometric. As a consequence, provided that the operation of a domain equation can be extend...

Morphisms from H to G in the Burnside category of groups are provided by sets with commuting right and left actions of H and G, which are free and finite over G. We extend this construction to groupoids by considering a G-action on a set X as a homomorphism G → Aut(X). This has an alternative expression in terms of “bi-actions,” a variant of the discrete version of a topological construction wh...

This is the 4rst of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of L-topological spaces. The authors begin by de4ning the notion of q-sets, which together with the p-sets considered earlier, enable the formulation of a powerful concept of duality. This plays an important...

To work effectively with simplicial sets, which provide useful algebraic or combinatorial models of topological spaces, we need to get serious about geometric realization, colimits, limits, and adjunctions. A first step is to recall the Yoneda Lemma. Given any pair of functors F,G : D−→C, denote by Nat(F,G) := {φ| φ : F−→G} the collection of all natural transformations from F to G. Denote by sS...

We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest, such as finite sets, Boolean functions, topological spaces, vector spaces, graded algeb...

The study of topological quantum field theories increasingly relies upon concepts from higherdimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a ‘suspension’ operation on n-categories, a...

The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a ‘suspension’ operation on ncategories, a...