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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