A New Category? Domains, Spaces and Equivalence Relations

The familiar categories SET and TOP, consisting of sets and arbitrary mappings and of topological spaces and continuous mappings, have many well known closure properties. For example, they are both complete and cocomplete, meaning that they have all (small) limits and colimits. They are well-powered and co-well-powered, meaning that collections of subobjects and quotients of objects can be represented by sets. They are also nicely related, since SET can be regarded as a full subcategory of TOP, and the forgetful functor that takes a topological space to its underlying set preserves limits and colimits (but reflects neither). The category SET is also a cartesian closed category, meaning that the function-space construct or the internal hom-functor is very well behaved. However, it has been known for a long time that in TOP no such assertion is available, because in general it is not possible to assign a topology to the set of continuous functions AB making this adjointness valid -- except under some special conditions on the space B.

The proposed solution to the problem of cartesian closedness is motivated by domain theory. The (new?) category is the the category of topological T0-spaces and arbitrary equivalence relations, to be called EQU, where the mappings are (suitable equivalence classes of) continuous mappings which preserve the equivalence relations. Let us call these spaces equilogical spaces and the mappings equivariant. It seems surprising that this category has not been noticed before -- if in fact it has not. It is easy to see that EQU is complete and cocomplete and that it embedds TOP0 as a full and faithful subcategory (by taking the equivalence relation to be the identity relation). What is perhaps not so obvious is that EQU is indeed cartesian closed. The proof of cartesian closedness uses old theorems in domain theory originally discovered by the author: in particular, an injective property of algebraic lattices treated as topological spaces and the fact that they form a cartesian closed category (along with continuous functions). The paper discusses other closure properties of EQU as well.

Table of Contents

1. Introduction
2. T0-Spaces and Algebraic Lattices
3. Equilogical Spaces
4. Some Subcategories
5. Acknowledgments and Questions