Structures such as braided monoidal categories, operads, and Hopf algebras are familiar to those who have studied topological quantum field theory, knot theory, string theory, and the renormalization procedure in quantum field theory. This book attempts, and succeeds, in presenting to the interested reader an overview of higher category theory, which subsumes the aforementioned topics. It is not however a book on applications, but instead details the purely mathematical aspects of higher category, clarifying for example the difference between `weak' n-categories and `strict' n-categories. The author though has not written a book in the typical "definition-theorem-proof" style, as he motivates the subject very well, and does not hesitate to use diagrams to get his point across. Indeed, he is careful to point out that the subject is inherently topological in its nature, and that diagrams used to illustrate higher-dimensional structures can be viewed as topological structures. The braided monoidal category that arises in knot theory is a perfect example of this.
The author introduces higher-dimensional category theory as one that uses "higher-dimensional arrows", in analogy to ordinary category as one that uses 1-dimensional arrows. Higher-dimensional category theory or `n-category theory,' is viewed as a generalization of the notion of category. To motivate the concept of a weak n-category, the author reminds the reader of the attempt to prove to what extent the loop group in differential topology is in fact a topological group. The composition of paths in the loop groups is not associative, but rather associative up to homotopy. Associativity does hold in strict n-categories but not in weak n-categories. As another example of non-associativity, the author discusses the fundamental omega-groupoid, which is the higher-dimensional category arising from a topological space. Several examples of weak n-categories are given in the motivating chapter of the book.
The first chapter is an overview of classical category theory, most of which may be review for readers familiar with it. Of particular importance is the notion of a monoidal category, which for the case of a strict monoidal category, generalizes the familiar tensor product operation. The tensor operation is generalized to that of a functor on a category that obeys strict associativity and unit laws. Weak monoidal categories are also defined, where the functor now obeys associativity and unit laws only up to isomorphism. These are the `coherence isomorphisms', and these satisfy the `pentagon' and `triangle axioms.' Modules over commuative rings with the usual tensor product are monoidal categories.
The author introduces operads in chapter two, concentrating first on multicategories, which are collections of objects on which are defined maps or "arrows", and compositions that satisfy associativity and unit properties. Operads are multicategories with only one object, and can be viewed as an abstraction of a set of composable functions of several variables where the variables can be permuted. Several examples are given of multicategories with many objects, including how a monoidal category can give rise to a multicategory. Operads appear in physical applications, such as string field theory and conformal field theory, which are not discussed in the book, but the author gives many examples of operads that make their properties readily apparent. One of these involves iterated loop spaces, where operads arose historically.
After a further discussion of monoidal categories in chapter 3, the author spends part two of the book solely on operads. One of the first goals of the author is formalize the notion of an input type, so as to allow more than just finite sequences of objects. For each input type he defines a theory of operads and multicategories, which yields the "plain" operad when the inputs are finite sequences. The author also discusses how to start with a monad T on any category and construct `T-multicategories'. T-operads are then T-multicategories with one object, and algebras can be associated to T-multicategories. These algebras are an analog of the "representation" or "model" for the T-multicategory. The author's work on "free category" or `fc-multicategories', which are 2-dimensional examples of these generalized multicategories. Fc-multicategories are T-multicategories on the free category (fc) monad on the category of directed graphs. As a very interesting example of an fc-multicategory, the author discusses one which encapsulates (in a single structure) rings, homomorphisms of rings, modules over rings, homomorphisms of modules, and tensor products of modules.
Also discussed in this part is the notion of an `opetope' (for "operation polytope"), which are a kind of generalization of the simplices of simplicial geometry. The opetopes are thus the "polytopes" of higher-dimensional category theory, and are defined by first taking for every natural number and defining a category and monad inductively. The zeroth category is Set and the zeroth monad is the identity. This gives rise to an infinite sequence of opetopes, with the zeroth opetope being 1. The nth category is then canonically isomorphic to Set modulo the nth opetope. A 2-opetope is the natural numbers, while a 3-opetope is the collection of trees. The author shows how to construct a category of n-dimensional pasting diagrams for each natural number n, where for n = 1 is the category of finite totally ordered sets, and for n = 2, the category of trees. The geometric connotations of the pasting diagrams are obvious, as well as their analogy to simplicial objects. An opetopic n-pasting diagram is defined as an (n+1)-opetope for each natural number n. 2-pasting diagrams correspond to trees, and the author shows how to construct `stable trees', i.e. those trees whose vertices have at least two branches coming out of them. The relation of stable trees to the constructions of Stasheff are discussed, along with the connection of opetopes to the construction of weak n-categories.
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Dr. Lee D. Carlson
A very interesting overview
2004年9月19日にアメリカ合衆国でレビュー済みAmazonで購入
