Higher-dimensional category

-category

Let be a natural number. An -category [a16] consists of sets , where the elements of are called -arrows and are, for all , equipped with a category structure for which is the set of objects and is the set of arrows, where the composition is denoted by (for composable ), such that, for all , there is a -category (cf. Bicategory) with , as set of objects, arrows and -arrows, respectively, with vertical composition , and with horizontal composition . The sets with the source and target functions form the underlying globular set (or -graph) of . For and for with the same -source and -target, there is an -category whose -arrows () are the -arrows of . In particular, for -arrows (also called objects), there is an -category . This provides the basis of an alternative definition [a17] of -category using recursion and enriched categories [a32] It follows that there is an -category -, whose objects are -categories and whose -arrows are -functors. For infinite , the notion of an -category [a44] is obtained. An -groupoid is an -category such that, for all , each -arrow is invertible with respect to the -composition (for infinite, -groupoid is used in [a9] rather than -groupoid, by which they mean something else).

One reason for studying -categories was to use them as coefficient objects for non-Abelian cohomology (cf. Cohomology). This required constructing the nerve of an -category which, in turn, required extending the notion of computad (cf. Bicategory) to -computad, defining free -categories on -computads, and formalising -pasting [a46]; [a22]; [a47]; [a23]; [a41].

Ever since the appearance of bicategories (i.e. weak -categories, cf. Bicategory) in 1967, the prospect of weak -categories () has been contemplated with some trepidation [a37], p. 1261. The need for monoidal bicategories arose in various contexts, especially in the theory of categories enriched in a bicategory [a53], where it was realized that a monoidal structure on the base was needed to extend results of usual enriched category theory [a32]. The general definition of a monoidal bicategory (as the one object case of a tricategory) was not published until [a19]; however, in 1985, the structure of a braiding [a26] was defined on a monoidal (i.e. tensor) category and was shown to be exactly what arose when a tensor product (independent of specific axioms) was present on the one-object bicategory . The connection between braidings and the Yang–Baxter equation was soon understood [a52], [a25]. This was followed by a connection between the Zamolodchikov equation and braided monoidal bicategories [a29], [a30] using more explicit descriptions of this last structure. The categorical formulation of tangles in terms of braiding plus adjunction (or duality; cf. also Adjunction theory) was then developed [a18]; [a45]; [a43]. See [a31] for the role this subject plays in the theory of quantum groups.

Not every tricategory is equivalent (in the appropriate sense) to a -category: the interchange law between - and -compositions needs to be weakened from an equality to an invertible coherent -cell; the groupoid case of this had arisen in unpublished work of A. Joyal and M. Tierney on algebraic homotopy -types in the early 1980s; details, together with the connection with loop spaces (cf. Loop space), can be found in [a8]; [a5]. (A different non-globular higher-groupoidal homotopy -type for all was established in [a35].) Whereas -categories are categories enriched in the category - of -categories with Cartesian product as tensor product, Gray categories (or "semi-strict 3-categories" ) are categories enriched in the monoidal category - where the tensor product is a pseudo-version of that defined in [a20]. The coherence theorem of [a19] states that every tricategory is (tri)equivalent to a Gray category. A basic example of a tricategory is whose objects are bicategories, whose arrows are pseudo-functors, whose -arrows are pseudo-natural transformations, and whose -arrows are modifications.

While a simplicial approach to defining weak -categories for all was suggested in [a46], the first precise definition was that of J. Baez and J. Dolan [a2] (announced at the end of 1995). Other, apparently quite different, definitions by M.A. Batanin [a6] and Z. Tamsamani [a50] were announced in 1996 and by A. Joyal [a24] in 1997. Both the Baez–Dolan and Batanin definitions involve different generalizations of the operads of P. May [a39] as somewhat foreshadowed by T. Trimble, whose operad approach to weak -categories had led to a definition of weak -category (or tetracategory) [a51].

With precise definitions available, the question of their equivalence is paramount. A modified version [a21] of the Baez–Dolan definition together with generalized computad techniques from [a7] are expected to show the equivalence of the Baez–Dolan and Batanin definitions.

The next problem is to find the correct coherence theorem for weak -categories: What are the appropriately stricter structures generalizing Gray categories for Strong candidates seem to be the "teisi" (Welsh for "stacks" ) of [a12], [a13], [a14]. Another problem is to find a precise definition of the weak -category of weak -categories.

The geometry of weak -categories () is only at its early stages [a40], [a18], [a33], [a3]; however, there are strong suggestions that this will lead to constructions of invariants for higher-dimensional manifolds and have application to conformal field theory [a10], [a1], [a11], [a36].

The theory of weak -categories, even for , is also in its infancy [a15], [a38]. Reasons for developing this theory, from the computer science viewpoint, are described in [a42]. There are applications to concurrent programming and term-rewriting systems; see [a48], [a49] for references.

References

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