Higher-dimensional category

$n$-category

Let $n$ be a natural number. An $n$-category $A$ [a16] consists of sets $A _ { 0 } , \ldots , A _ { n }$, where the elements of $A _ { m }$ are called $m$-arrows and are, for all $0 \leq k < m \leq n$, equipped with a category structure for which $A _ { k }$ is the set of objects and $A _ { m }$ is the set of arrows, where the composition is denoted by $a \circ_{k} b$ (for composable $a , b \in A _ { m }$), such that, for all $0 \leq h < k < m \leq n$, there is a $2$-category (cf. Bicategory) with $A _ { h } , A _ { k } , A _ { m }$, as set of objects, arrows and $2$-arrows, respectively, with vertical composition $a \circ_{k} b$, and with horizontal composition $a \circ_{h} b$. The sets $A _ { m }$ with the source and target functions $A _ { m } \rightarrow A _ { m - 1 }$ form the underlying globular set (or $n$-graph) of $A$. For $0 \leq k \leq n$ and for $a , b \in A _ { k }$ with the same $( k - 1 )$-source and $( k - 1 )$-target, there is an $( n - k - 1 )$-category $A ( a , b )$ whose $m$-arrows ($k < m \leq n$) are the $m$-arrows $c : a \rightarrow b$ of $A$. In particular, for $0$-arrows $a , b$ (also called objects), there is an $( n + 1 )$-category $A ( a , b )$. This provides the basis of an alternative definition [a17] of $n$-category using recursion and enriched categories [a32] It follows that there is an $(n + 1)$-category $n$-Cat, whose objects are $n$-categories and whose $1$-arrows are $n$-functors. For infinite $n$, the notion of an $\omega$-category [a44] is obtained. An $n$-groupoid is an $n$-category such that, for all $0 < m \leq n$, each $m$-arrow is invertible with respect to the $( m - 1 )$-composition (for $n$ infinite, $\infty$-groupoid is used in [a9] rather than $\omega$-groupoid, by which they mean something else).

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

Ever since the appearance of bicategories (i.e. weak $2$-categories, cf. Bicategory) in 1967, the prospect of weak $n$-categories ($n > 2$) 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 $\mathcal{V}$ and was shown to be exactly what arose when a tensor product (independent of specific axioms) was present on the one-object bicategory $\Sigma \cal V$. 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 $3$-category: the interchange law between $0$- and $1$-compositions needs to be weakened from an equality to an invertible coherent $3$-cell; the groupoid case of this had arisen in unpublished work of A. Joyal and M. Tierney on algebraic homotopy $3$-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 $n$-type for all $n$ was established in [a35].) Whereas $3$-categories are categories enriched in the category $2$-Cat of $2$-categories with Cartesian product as tensor product, Gray categories (or "semi-strict 3-categories" ) are categories enriched in the monoidal category $2$-Cat 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 BiCat whose objects are bicategories, whose arrows are pseudo-functors, whose $2$-arrows are pseudo-natural transformations, and whose $3$-arrows are modifications.

While a simplicial approach to defining weak $n$-categories for all $n$ 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 $n$-categories had led to a definition of weak $4$-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 $n$-categories: What are the appropriately stricter structures generalizing Gray categories for $n = 3 ?$ 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 $( n + 1 )$-category of weak $n$-categories.

The geometry of weak $n$-categories ($n > 2$) 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 $n$-categories, even for $n = 3$, 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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