Epilogue
Published online by Cambridge University Press: 25 October 2011
Summary
We have given a thorough treatment of some of the main first elements of the theory of higher categories, adopting Segal's method of weak enrichment in an iterative way. The fundamental step is the construction of the cartesian model category PC(M) in a way which can be iterated to give PCn(M). The case of n-categories is obtained by starting with M = Set. The case of Segal n-categories, which are (∞, n)-categories in Lurie's terminology, is obtained by starting with the standard model category K of simplicial sets.
The internal Hom within one of these iteratively constructed cartesian model categories P provides the morphism spaces which go together to create a P-enriched category Enr(P). For P = PCn(Set) this gives nCAT, the (n + 1)-category of weakly associative n-categories. For P = PCn(K) this gives nSeCAT, the (∞, n + 1)-category of (∞, n)-categories.
In the last part of the book, we have indicated how to start towards the development of various aspects of the theory of higher categories using this model structure. This is only a start, touching the questions of inverting morphisms and localization, and limits and colimits. It leaves open a vast expanse of questions, many of which are the subject of recent and ongoing research by many people.
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- Homotopy Theory of Higher CategoriesFrom Segal Categories to n-Categories and Beyond, pp. 616 - 617Publisher: Cambridge University PressPrint publication year: 2011