[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

*To*: categories@mta.ca*Subject*: categories: Thesis: `Operads in higher-dimensional category theory'*From*: Tom Leinster <T.Leinster@dpmms.cam.ac.uk>*Date*: Fri, 8 Dec 2000 01:38:50 +0000 (GMT)*Sender*: cat-dist@mta.ca

My PhD thesis, `Operads in higher-dimensional category theory', is now (approved and) available electronically as math.CT/0011106 - that is, at http://arXiv.org/abs/math.CT/0011106 The summary follows. Tom Operads in Higher-Dimensional Category Theory Tom Leinster The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak omega-category. This theory of operads and multicategories has various other applications too: for instance, to the opetopic approach to n-categories expounded by Baez, Dolan and others, and to the theory of enrichment of higher-dimensional categorical structures. We sketch some of these further developments, without exploring them in full. We start with a look at bicategories (Chapter 1). Having reviewed the basics of the classical definition, we define `unbiased bicategories', in which n-fold composites of 1-cells are specified for all natural n (rather than the usual nullary and binary presentation). We go on to show that the theories of (classical) bicategories and of unbiased bicategories are equivalent, in a strong sense. The heart of this work is the theory of generalized operads and multicategories. More exactly, given a monad T on a category E, satisfying simple conditions, there is a theory of T-operads and T-multicategories. In Chapter 2 we set up the basic concepts of the theory, including the important definition of an algebra for a T-multicategory. In Chapter 3 we cover an assortment of further operadic topics, some of which are used in later parts of the thesis, and some of which pertain to the applications mentioned in the first paragraph. Chapter 4 is a (proposed) definition of weak omega-category, a modification of that given by Batanin (Adv Math 136 (1998), 39-103). Having given the definition formally, we take a long look at why it is a *reasonable* definition. We then explore weak n-categories (for finite n), and show that weak 2-categories are exactly unbiased bicategories. The four appendices take care of various details which would have been distracting in the main text. Appendix A contains the proof that unbiased bicategories are essentially the same as classical bicategories. Appendix B describes how to form the free T-multicategory on a given T-graph. In Appendix C we discuss various facts about strict omega-categories, including a proof that the category they form is monadic over an appropriate category of graphs. Finally, Appendix D is a proof of the existence of an initial object in a certain category, as required in Chapter 4.

- Prev by Date:
**categories: Re: Can we ignore smallness?** - Next by Date:
**categories: Terminology** - Prev by thread:
**categories: Re: Can we ignore smallness?** - Next by thread:
**categories: final functors** - Index(es):