# 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 $n$-, 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 $2$- of $2$-categories with Cartesian product as tensor product, Gray categories (or "semi-strict 3-categories" ) are categories enriched in the monoidal category $2$- 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 $2$-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

[a1] | J. Baez, J. Dolan, "Higher-dimensional algebra and topological quantum field theory" J. Math. Phys. , 36 (1995) pp. 6073–6105 |

[a2] | J. Baez, J. Dolan, "Higher-dimensional algebra III: $n$-categories and the algebra of opetopes" Adv. Math. , 135 (1998) pp. 145–206 |

[a3] | J. Baez, L. Langford, "Higher-dimensional algebra IV: $2$-tangles" http://math.ucr.edu/home/baez/hda4.ps (1999) |

[a4] | J. Baez, M. Neuchl, "Higher-dimensional algebra I: braided monoidal $2$-categories" Adv. Math. , 121 (1996) pp. 196–244 |

[a5] | C. Balteanu, Z. Fierderowicz, R. Schwaenzl, R. Vogt, "Iterated monoidal categories" Preprint Ohio State Math. Research Inst. , 5 (1998) |

[a6] | M.A. Batanin, "Monoidal globular categories as natural environment for the theory of weak $n$-categories" Adv. Math. , 136 (1998) pp. 39–103 |

[a7] | M.A. Batanin, "Computads for finitary monads on globular sets" , Higher Category Theory (Evanston, Ill, 1997) , Contemp. Math. , 230 , Amer. Math. Soc. (1998) pp. 37–57 |

[a8] | C. Berger, "Double loop spaces, braided monoidal categories and algebraic $3$-types of space" Prépubl. Univ. Nice-Sophia Antipolis, Lab. Jean-Alexandre Dieudonné , 491 (1997) |

[a9] | R. Brown, P.J. Higgins, "The equivalence of crossed complexes and $\infty$-groupoids" Cah. Topol. Géom. Diff. Cat. , 22 (1981) pp. 371–386 |

[a10] | S.M. Carmody, "Cobordism categories" PhD Thesis Univ. Cambridge (1995) |

[a11] | L. Crane, D.N. Yetter, "A categorical construction of $4 D$ topological quantum field theories" L.H. Kauffman (ed.) R.A. Baadhio (ed.) , Quantum Topology , World Sci. (1993) pp. 131–138 |

[a12] | S. Crans, "Generalized centers of braided and sylleptic monoidal $2$-categories" Adv. Math. , 136 (1998) pp. 183–223 |

[a13] | S. Crans, "A tensor product for Gray-categories" Theory Appl. Categ. , 5 (1999) pp. 12–69 |

[a14] | S. Crans, "On braidings, syllepses, and symmetries" Cah. Topol. Géom. Diff. Cat. (to appear) |

[a15] | B.J. Day, R. Street, "Monoidal bicategories and Hopf algebroids" Adv. Math. , 129 (1997) pp. 99–157 |

[a16] | C. Ehresmann, "Catégories et structures" , Dunod (1965) |

[a17] | S. Eilenberg, G.M. Kelly, "Closed categories" , Proc. Conf. Categorical Algebra, La Jolla , Springer (1966) pp. 421–562 |

[a18] | J. Fischer, "2-categories and 2-knots" Duke Math. J. , 75 (1994) pp. 493–526 |

[a19] | R. Gordon, A.J. Power, R. Street, "Coherence for tricategories" Memoirs Amer. Math. Soc. , 117 : 558 (1995) |

[a20] | J.W. Gray, "Coherence for the tensor product of $2$-categories, and braid groups" , Algebra, Topology, and Category Theory (a collection of papers in honour of Samuel Eilenberg) , Acad. Press (1976) pp. 63–76 |

[a21] | C. Hermida, M. Makkai, J. Power, "On weak higher dimensional categories" , http://hypatia.dcs.qmw.ac.uk/authors/M/MakkaiM/papers/multitopicsets/ (1999) |

[a22] | M. Johnson, "Pasting diagrams in $n$-categories with applications to coherence theorems and categories of paths" PhD Thesis Univ. Sydney, Australia (1987) |

[a23] | M. Johnson, "The combinatorics of $n$-categorical pasting" J. Pure Appl. Algebra , 62 (1989) pp. 211–225 |

[a24] | A. Joyal, "Disks, duality and $\Theta$-categories" Preprint and Talk at the Amer. Math. Soc. Meeting in Montréal, September (1997) |

[a25] | A. Joyal, R. Street, "Tortile Yang–Baxter operators in tensor categories" J. Pure Appl. Algebra , 71 (1991) pp. 43–51 |

[a26] | A. Joyal, R. Street, "Braided tensor categories" Adv. Math. , 102 (1993) pp. 20–78 |

[a27] | M.M. Kapranov, V.A. Voevodsky, "Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (List of results)" Cah. Topol. Géom. Diff. Cat. , 32 (1991) pp. 11–27 |

[a28] | M.M. Kapranov, V.A. Voevodsky, "Groupoids and homotopy types" Cah. Topol. Géom. Diff. Cat. , 32 (1991) pp. 29–46 |

[a29] | M.M. Kapranov, V.A. Voevodsky, "$2$-Categories and Zamolodchikov tetrahedra equations" , Proc. Symp. Pure Math. , 56 , Amer. Math. Soc. (1994) pp. 177–259 |

[a30] | M.M. Kapranov, V.A. Voevodsky, "Braided monoidal $2$-categories and Manin–Schechtman higher braid groups" J. Pure Appl. Algebra , 92 (1994) pp. 241–267 |

[a31] | C. Kassel, "Quantum groups" , Graduate Texts Math. : 155 , Springer (1995) |

[a32] | G.M. Kelly, "Basic concepts of enriched category theory" , Lecture Notes London Math. Soc. : 64 , Cambridge Univ. Press (1982) |

[a33] | V. Kharlamov, V. Turaev, "On the definition of the $2$-category of $2$-knots" Transl. Amer. Math. Soc. , 174 (1996) pp. 205–221 |

[a34] | L. Langford, "$2$-Tangles as a free braided monoidal $2$-category with duals" PhD Thesis Univ. California at Riverside (1997) |

[a35] | J.-L. Loday, "Spaces with finitely many non-trivial homotopy groups" J. Pure Appl. Algebra , 24 (1982) pp. 179–202 |

[a36] | M. Mackay, "Spherical $2$-categories and $4$-manifold invariants" Adv. Math. , 143 (1999) pp. 288–348 |

[a37] | S. MacLane, "Possible programs for categorists" , Lecture Notes Math. , 86 , Springer (1969) pp. 123–131 |

[a38] | F. Marmolejo, "Distributive laws for pseudomonads" Theory Appl. Categ. , 5 (1999) pp. 91–147 |

[a39] | P. May, "The geometry of iterated loop spaces" , Lecture Notes Math. , 271 , Springer (1972) |

[a40] | M. McIntyre, T. Trimble, "The geometry of Gray-categories" Adv. Math. (to appear) |

[a41] | A.J. Power, "An $n$-categorical pasting theorem" A. Carboni (ed.) M.C. Pedicchio (ed.) G. Rosolini (ed.) , Category Theory, Proc. Como 1990 , Lecture Notes Math. , 1488 , Springer (1991) pp. 326–358 |

[a42] | A.J. Power, "Why tricategories?" Inform. Comput. , 120 (1995) pp. 251–262 |

[a43] | N.Yu. Reshetikhin, V.G. Turaev, "Ribbon graphs and their invariants derived from quantum groups" Comm. Math. Phys. , 127 (1990) pp. 1–26 |

[a44] | J.E. Roberts, "Mathematical aspects of local cohomology" , Proc. Colloq. Operator Algebras and Their Application to Math. Physics, Marseille 1977 , CNRS (1979) |

[a45] | M.C. Shum, "Tortile tensor categories" J. Pure Appl. Algebra , 93 (1994) pp. 57–110 (PhD Thesis Macquarie Univ. Nov. 1989) |

[a46] | R. Street, "The algebra of oriented simplexes" J. Pure Appl. Algebra , 49 (1987) pp. 283–335 |

[a47] | R. Street, "Parity complexes" Cah. Topol. Géom. Diff. Cat. , 32 (1991) pp. 315–343 (Corrigenda: 35 (1994) 359-361) |

[a48] | R. Street, "Categorical structures" M. Hazewinkel (ed.) , Handbook of Algebra , I , Elsevier (1996) pp. 529–577 |

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[a50] | Z. Tamsamani, "Sur des notions de $n$-categorie et $n$-groupoide non-stricte via des ensembles multi-simpliciaux" PhD Thesis Univ. Paul Sabatier, Toulouse (1996) (Also available on alg-geom 95-12 and 96-07) |

[a51] | T. Trimble, "The definition of tetracategory" Handwritten diagrams, , August (1995) |

[a52] | V.G. Turaev, "The Yang–Baxter equation and invariants of links" Invent. Math. , 92 (1988) pp. 527–553 |

[a53] | R.F.C. Walters, "Sheaves on sites as Cauchy-complete categories" J. Pure Appl. Algebra , 24 (1982) pp. 95–102 |

**How to Cite This Entry:**

Higher-dimensional category.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Higher-dimensional_category&oldid=50809