Changes between arXiv version 2 and arXiv version 3
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(These are changes in addition to the items in the referee report.)

Many small clarifications of text.  

Many typos corrected.


The most significant changes are:

- added to acknowledgements
- clarified definition of splittable
- changed to pitchfork notation for splittable subsets of fields
- added definition of collaring homeomorphism
- improved definition of bordism n-category
- fixed definition of a refinement of a ball decomposition (intermediate manifolds should also be disjoint unions of balls)
- added brief definition of monoidal n-categories
- fixed statement of compatibility of product morphisms with gluing
- added remark about manifolds which do not admit ball decompositions; restricted product theorem (7.1.1) to apply only to these manifolds
- added remarks about categories of defects
- clarified that the "cell complexes" in string diagrams are actually a bit more general
- added remark to insure that the poset of decompositions is a small category
- corrected statement of module to category restrictions
- reduced intermingling for the various n-cat definitions (plain, enriched, A-infinity)
- strengthened n-cat isotopy invariance axiom to allow for homeomorphisms which act trivially elements on the restriction of an n-morphism to the boundary of the ball
- more details on axioms for enriched n-cats
- added details to the construction of traditional 1-categories from disklike 1-categories (Appendix C.1)
- extended the lemmas of Appendix B (about adapting families of homeomorphisms to open covers) to the topological category
- modified families-of-homeomorphisms-action axiom for A-infinity n-categories, and added discussion of alternatives
- added n-cat axiom for existence of splittings, and added similar axiom for fields
- added transversality requirement to product morphism axiom
- added remarks on Morita equivalence for n-categories
- rewrote definition of colimit (in "From Balls to Manifolds" subsection) to allow for more general decompositions; also added more details
- added remark about families of collar maps acting on the blob complex
- small corrections to proof of product theorem (7.1.1)
- added remarks that various homotopy equivalences we construct are well-defined up to a contractible set of choices
- clarified that the surgery cylinder operad action is only up to coherent homotopy
- added some details to the construction of a traditional 2-category from a disk-like 2-category