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Most of my work is in higher-dimensional category theory. If you want an
introduction to this subject then you could look at
John Baez's
website, or the detailed research proposal
(dvi,
pdf)
on
Peter May's
page, or my paper
Topology and
higher-dimensional category theory: the rough idea.
My latest work is on self-similarity. There's nothing on this page about that yet, but I have written an overview paper. For the purposes of this guide I've divided my papers into five groups:
All links to papers open in a new window. New! My book, Higher Operads, Higher Categories, supersedes the papers listed in the first two sections, 'Operads and multicategories' and 'n-categories'. |
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To many people, 'higher-dimensional category theory' means the study of
n-categories. But I'll start by describing my work on some other
very interesting higher-dimensional categorical structures: generalized
operads and multicategories. Amongst other things, these turn out to
provide a very natural and flexible language in which n-categories
can be discussed.
A category is made up of arrows like a ----> b, where the domain is a single object and the codomain is a single object. A multicategory is like a category, but has arrows like (a1, a2, ......, ak) ----> b where the domain is a finite sequence of objects (and the codomain is still a single object). Think of an arrow as a box with k input wires coming in on the left and one output wire emerging on the right. With this in mind, it's easy to imagine what composition in a multicategory looks like. A typical multicategory is that of vector spaces and multilinear maps. Those are ordinary multicategories. In the theory of generalized multicategories, the shape of the domain of an arrow can be something more exotic than a single object or a sequence of objects: it might, for instance, be a tree of objects, a d-dimensional array of objects, or a diagram of pasted-together cells. You may be more familiar with operads than multicategories. Formally, an operad is just a multicategory with only one object. So multicategories are to operads as categories are to monoids. Some people call multicategories 'coloured operads', and similarly you might call categories 'coloured monoids' and groupoids 'coloured groups'. Operads can be thought of as algebraic theories of a certain kind. (But really, Are operads algebraic theories?) By using generalized operads, we capture the kind of algebraic theory that the theory of n-categories is. The point is that for ordinary operads, the input of an operation is a one-dimensional sequence of things, but with generalized operads we can describe operations that take as input a higher-dimensional array of things. For instance, composition in an n-category will be an operation of this kind. A formal definition of generalized multicategory (and generalized operad), along with some of their theory, is in General operads and multicategories. Similar accounts can be found in Operads in higher-dimensional category theory (probably the best source) and Structures in higher-dimensional category theory, both of which are described more fully under 'n-Categories' next. |
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Many definitions of (weak) n-category have been proposed. Ten such
definitions are given, in two pages each, in
A survey of definitions of n-category.
This also contains, for each of the definitions, a two-page analysis of the
cases n = 0,1,2, and a bibliography with commentary.
Several of these proposed definitions can be phrased neatly in the language of generalized operads. I have investigated one of them particularly carefully, using this language, in two different papers. The better account is in Operads in higher-dimensional category theory, which contains
An earlier paper, Structures in higher-dimensional category theory, also covers the first two of these three areas, although more crudely. It also contains
Mostly for my own convenience, I wrote out the definitions of (classical) bicategory and of functor, transformation and modification between bicategories, together with a bare-bones account of the coherence theorem: Basic bicategories. |
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If you want to teach a student what a 'category enriched in abelian groups'
(= 'Ab-category') is, then it's entirely unnecessary for them to
know about tensor products: it's enough that they just know the
definition of multilinear map. In other words, there's really no such
thing as a 'category enriched in a monoidal category': it's actually the
underlying multicategory in which you enrich.
This leads to the general question: given some kind of categorical structure, in what kind of categorical structure might it be enriched? In the previous paragraph, the given categorical structure was just a plain category, and so the question was: 'what kind of thing must V be if we are to speak sensibly of V-enriched categories?' There is a precise and surprising answer to this question when the given categorical structures are generalized multicategories. It turns out that if we take generalized multicategories where the arrows have input shape of one kind, then the structures in which these can be enriched are generalized multicategories where the input shapes are of a more complicated kind. (See Operads and Multicategories above for what I mean by the 'input shape' of an arrow.) For example, we've just seen that categories - which are generalized multicategories where the input (domain) of an arrow is a single object - are capable of being enriched in ordinary multicategories - which are generalized multicategories where the input of an arrow is a sequence of objects. All this is done in Generalized enrichment for categories and multicategories. There are a couple of applications. Firstly, we see how the operads that often arise in mathematics, where the set of k-ary operations comes equipped with the structure of a space or a module or something else, fit into our theory of generalized operads and multicategories. Secondly, we show how 'relaxed multilinear categories' (certain structures that arose independently in two different branches of quantum algebra) can be seen as a very natural example of enriched generalized multicategories. Categories are a special case of generalized multicategories, so the answer to 'what can a generalized multicategory be enriched in?' specializes to tell us what a category can be enriched in. From what's been said already you might expect the answer to be 'an ordinary multicategory', but in fact it's something even more general: a so-called 'fc-multicategory'. These are very general two-dimensional structures, encompassing monoidal categories, ordinary multicategories, bicategories, double categories and more. The theory of categories enriched in an fc-multicategory is explained as part of the general theory in Generalized enrichment for categories and multicategories, but there are also two direct accounts, not requiring knowledge of generalized multicategories: fc-multicategories, and the more detailed Generalized enrichment of categories. |
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Why are there are so many proposed definitions of n-category, and
why does no-one know how they're related? What makes it so hard? One of
the main problems is that it's weak n-categories we're trying
to define, where 'weak' means that laws such as associativity only hold
up to equivalence in some coherent way. Understanding exactly what
this should or might mean, and being able to reason about the whole process
of weakening, seems far from straightforward.
There is a closely analogous problem in topology. Often one encounters topological spaces that are naturally equipped with an almost-algebraic structure: one has the operations, but the axioms are only satisfied up to homotopy. For example, a loop space (the space of paths from a fixed basepoint to itself in some fixed space) is an 'up-to-homotopy topological group' under the operation of composition of loops; it's not a genuine group, because laws such as associativity don't hold exactly. In this example and many others, one can choose particular homotopies to witness that the laws hold up to homotopy, and then these homotopies obey further laws up to homotopy, and so on. So the full situation is quite complicated. As for n-categories, there are various different precise formulations of 'up-to-homotopy algebraic structure', although here more is known about how the different formulations compare. My own contribution has been to give a definition of up-to-homotopy algebraic structure that is both simple and valid in a very general context. More exactly, the definition is of a homotopy algebra in M for any operad, where M is any monoidal category equipped with a class of morphisms called 'homotopy equivalences'. So, for instance, it gives us a notion of homotopy differential graded algebra for an operad. (The idea is a generalization of the Gamma-space idea of Graeme Segal, Gamma-spaces being homotopy commutative monoids.) The full-length account is Homotopy algebras for operads. It has to be said that there are many loose ends and puzzling aspects to this theory. A short and less perplexing account, which does not assume familiarity with operads but still ought to convey the main idea, is Up-to-homotopy monoids. |
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Topology and higher-dimensional category theory: the rough idea
is a set of notes from informal talks I gave to some other mathematicians,
mostly geometers and topologists, trying to explain some higher-dimensional
category theory to them. The focus is on the many links with topology: for
instance, the links with homotopy algebra mentioned
above,
the idea that tame topology should be the study of n-groupoids, the
hope that higher categorical structures will provide a clean setting for
TQFT, and the connections between n-categories and homotopy groups
of spheres.
Categories are often like systems of numbers: the objects can be added, multiplied, and so on. This analogy is beneficial in both directions. If a system of numbers can be realized as the set of objects of a category then it becomes more meaningful - one has a notion of a map between numbers, for instance. An objective representation of the Gaussian integers is an example of this. Conversely, it is sometimes valid to do apparently illegal manipulations of objects of categories, pretending that they are ordinary numbers: Objects of categories as complex numbers. This is all joint work with Marcelo Fiore. Finally, a not-very-serious article I wrote a long time ago: Perfect numbers and groups generalizes the concept of a number being perfect (that is, equal to the sum of its proper divisors) to give a concept of a group being perfect (that is, its order is the sum of the orders of its proper normal subgroups). So perfect cyclic groups are the same as perfect numbers. It's shown that in fact these are the only perfect abelian groups, but it's also shown that there are examples of (even-order) perfect nonabelian groups. All this has nothing to do with the usual meaning of 'perfect group', or, probably, much else in life.
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