Title:

The ωcategories associated with products of infinitedimensional globes

The results in this thesis are organised in four chapters. Chapter 1 is preliminary. We state the necessary definitions and results in ocomplexes, atomic complexes and products of ocomplexes. Some definitions are restated to meet the requirement for the following chapters. There is a new proof for the existence of 'natural homomorphism' (Theorem 1.3.6) and a new result for the decomposition of molecules in loopfree ocomplexes (Theorem 1.4.13). In Chapter 2, we study the product of three infinite dimensional globes. The main result in this chapter is that a subcomplex in the product of three infinite dimensional globes is a molecule if and only if it is pairwise molecular (Theorem 2.1.6). The definition for pairwise molecular subcomplexes is given in section 1. One direction of the main theorem, molecules are necessarily pairwise molecular, is proved in section 2. Some properties of pairwise molecular subcomplexes are studied in section 3. These properties are the preparation for a more explicit description of pairwise molecular subcomplexes, which is given in section 4. The properties for the sources and targets of pairwise molecular subcomplexes are studied in section 5, where we prove that the class of pairwise molecular subcomplexes is closed under source and target operation; there are also algorithms to calculate the sources and targets of a pairwise molecular subcomplex. Section 6 deals with the composition of pairwise molecular subcomplexes. The proof of the main theorem is completed in section 7, where an algorithm for decomposing molecules into atoms is implied in the proof. The construction of molecules in the product of three infinite dimensional globes is studied in Chapter 3. The main result is that any molecule can be constructed inductively by a systematic approach. Section 1 gives another description for molecules in the product of three infinite dimensional globes which is the theoretical basis for the construction. Section 2 states the inductive process of constructing molecules. The justification for the construction is given in section 3. The main result in Chapter 4 is that a subcomplex in the product of four infinite dimensional globes is a molecule if and only if it is pairwise molecular (Theorem 4.1.4). In the first four sections, some basic concepts and properties have to be reestablished to suit more general case. The organisation for the last three sections is parallel to that in Chapter 2. The corresponding results for sources, targets, composition and decomposition of pairwise molecular subcomplexes are also achieved.
