Difference between revisions of "Higher-dimensional category"
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+ | ''$n$-category'' | ||
− | + | Let $n$ be a natural number. An $n$-category $A$ [[#References|[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|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(2)|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 [[#References|[a17]]] of $n$-category using recursion and enriched categories [[#References|[a32]]] It follows that there is an $( n + 1 )$-category $n$-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120070/h12007045.png"/>, whose objects are $n$-categories and whose $1$-arrows are $n$-functors. For infinite $n$, the notion of an $\omega$-category [[#References|[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 [[#References|[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|Cohomology]]). This required constructing the nerve of an $n$-category which, in turn, required extending the notion of computad (cf. [[Bicategory(2)|Bicategory]]) to $n$-computad, defining free $n$-categories on $n$-computads, and formalising $n$-pasting [[#References|[a46]]]; [[#References|[a22]]]; [[#References|[a47]]]; [[#References|[a23]]]; [[#References|[a41]]]. | |
− | + | Ever since the appearance of bicategories (i.e. weak $2$-categories, cf. [[Bicategory(2)|Bicategory]]) in 1967, the prospect of weak $n$-categories ($n > 2$) has been contemplated with some trepidation [[#References|[a37]]], p. 1261. The need for monoidal bicategories arose in various contexts, especially in the theory of categories enriched in a bicategory [[#References|[a53]]], where it was realized that a monoidal structure on the base was needed to extend results of usual enriched category theory [[#References|[a32]]]. The general definition of a monoidal bicategory (as the one object case of a tricategory) was not published until [[#References|[a19]]]; however, in 1985, the structure of a braiding [[#References|[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|Yang–Baxter equation]] was soon understood [[#References|[a52]]], [[#References|[a25]]]. This was followed by a connection between the Zamolodchikov equation and braided monoidal bicategories [[#References|[a29]]], [[#References|[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|Adjunction theory]]) was then developed [[#References|[a18]]]; [[#References|[a45]]]; [[#References|[a43]]]. See [[#References|[a31]]] for the role this subject plays in the theory of [[Quantum groups|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|Loop space]]), can be found in [[#References|[a8]]]; [[#References|[a5]]]. (A different non-globular higher-groupoidal homotopy $n$-type for all $n$ was established in [[#References|[a35]]].) Whereas $3$-categories are categories enriched in the category $2$-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120070/h12007093.png"/> of $2$-categories with Cartesian product as tensor product, Gray categories (or "semi-strict 3-categories" ) are categories enriched in the monoidal category $2$-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120070/h12007097.png"/> where the tensor product is a pseudo-version of that defined in [[#References|[a20]]]. The coherence theorem of [[#References|[a19]]] states that every tricategory is (tri)equivalent to a Gray category. A basic example of a tricategory is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120070/h12007098.png"/> 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 [[#References|[a46]]], the first precise definition was that of J. Baez and J. Dolan [[#References|[a2]]] (announced at the end of 1995). Other, apparently quite different, definitions by M.A. Batanin [[#References|[a6]]] and Z. Tamsamani [[#References|[a50]]] were announced in 1996 and by A. Joyal [[#References|[a24]]] in 1997. Both the Baez–Dolan and Batanin definitions involve different generalizations of the operads of P. May [[#References|[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) [[#References|[a51]]]. | ||
With precise definitions available, the question of their equivalence is paramount. A modified version [[#References|[a21]]] of the Baez–Dolan definition together with generalized computad techniques from [[#References|[a7]]] are expected to show the equivalence of the Baez–Dolan and Batanin definitions. | With precise definitions available, the question of their equivalence is paramount. A modified version [[#References|[a21]]] of the Baez–Dolan definition together with generalized computad techniques from [[#References|[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 | + | 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 [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]]. Another problem is to find a precise definition of the weak $( n + 1 )$-category of weak $n$-categories. |
− | The geometry of weak | + | The geometry of weak $n$-categories ($n > 2$) is only at its early stages [[#References|[a40]]], [[#References|[a18]]], [[#References|[a33]]], [[#References|[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 [[#References|[a10]]], [[#References|[a1]]], [[#References|[a11]]], [[#References|[a36]]]. |
− | The theory of weak | + | The theory of weak $n$-categories, even for $n = 3$, is also in its infancy [[#References|[a15]]], [[#References|[a38]]]. Reasons for developing this theory, from the computer science viewpoint, are described in [[#References|[a42]]]. There are applications to concurrent programming and term-rewriting systems; see [[#References|[a48]]], [[#References|[a49]]] for references. |
====References==== | ====References==== | ||
− | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> J. Baez, J. Dolan, "Higher-dimensional algebra and topological quantum field theory" ''J. Math. Phys.'' , '''36''' (1995) pp. 6073–6105</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> J. Baez, J. Dolan, "Higher-dimensional algebra III: $n$-categories and the algebra of opetopes" ''Adv. Math.'' , '''135''' (1998) pp. 145–206</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J. Baez, L. Langford, "Higher-dimensional algebra IV: $2$-tangles" ''http://math.ucr.edu/home/baez/hda4.ps'' (1999)</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> J. Baez, M. Neuchl, "Higher-dimensional algebra I: braided monoidal $2$-categories" ''Adv. Math.'' , '''121''' (1996) pp. 196–244</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> C. Balteanu, Z. Fierderowicz, R. Schwaenzl, R. Vogt, "Iterated monoidal categories" ''Preprint Ohio State Math. Research Inst.'' , '''5''' (1998)</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> M.A. Batanin, "Monoidal globular categories as natural environment for the theory of weak $n$-categories" ''Adv. Math.'' , '''136''' (1998) pp. 39–103</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> 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)</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> R. Brown, P.J. Higgins, "The equivalence of crossed complexes and $\infty$-groupoids" ''Cah. Topol. Géom. Diff. Cat.'' , '''22''' (1981) pp. 371–386</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> S.M. Carmody, "Cobordism categories" ''PhD Thesis Univ. Cambridge'' (1995)</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a12]</td> <td valign="top"> S. Crans, "Generalized centers of braided and sylleptic monoidal $2$-categories" ''Adv. Math.'' , '''136''' (1998) pp. 183–223</td></tr><tr><td valign="top">[a13]</td> <td valign="top"> S. Crans, "A tensor product for Gray-categories" ''Theory Appl. Categ.'' , '''5''' (1999) pp. 12–69</td></tr><tr><td valign="top">[a14]</td> <td valign="top"> S. Crans, "On braidings, syllepses, and symmetries" ''Cah. Topol. Géom. Diff. Cat.'' (to appear)</td></tr><tr><td valign="top">[a15]</td> <td valign="top"> B.J. Day, R. Street, "Monoidal bicategories and Hopf algebroids" ''Adv. Math.'' , '''129''' (1997) pp. 99–157</td></tr><tr><td valign="top">[a16]</td> <td valign="top"> C. Ehresmann, "Catégories et structures" , Dunod (1965)</td></tr><tr><td valign="top">[a17]</td> <td valign="top"> S. Eilenberg, G.M. Kelly, "Closed categories" , ''Proc. Conf. Categorical Algebra, La Jolla'' , Springer (1966) pp. 421–562</td></tr><tr><td valign="top">[a18]</td> <td valign="top"> J. Fischer, "2-categories and 2-knots" ''Duke Math. J.'' , '''75''' (1994) pp. 493–526</td></tr><tr><td valign="top">[a19]</td> <td valign="top"> R. Gordon, A.J. Power, R. Street, "Coherence for tricategories" ''Memoirs Amer. Math. Soc.'' , '''117''' : 558 (1995)</td></tr><tr><td valign="top">[a20]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a21]</td> <td valign="top"> C. Hermida, M. Makkai, J. Power, "On weak higher dimensional categories" , http://hypatia.dcs.qmw.ac.uk/authors/M/MakkaiM/papers/multitopicsets/ (1999)</td></tr><tr><td valign="top">[a22]</td> <td valign="top"> M. Johnson, "Pasting diagrams in $n$-categories with applications to coherence theorems and categories of paths" ''PhD Thesis Univ. Sydney, Australia'' (1987)</td></tr><tr><td valign="top">[a23]</td> <td valign="top"> M. Johnson, "The combinatorics of $n$-categorical pasting" ''J. Pure Appl. Algebra'' , '''62''' (1989) pp. 211–225</td></tr><tr><td valign="top">[a24]</td> <td valign="top"> A. Joyal, "Disks, duality and $\Theta$-categories" ''Preprint and Talk at the Amer. Math. Soc. Meeting in Montréal, September'' (1997)</td></tr><tr><td valign="top">[a25]</td> <td valign="top"> A. Joyal, R. Street, "Tortile Yang–Baxter operators in tensor categories" ''J. Pure Appl. Algebra'' , '''71''' (1991) pp. 43–51</td></tr><tr><td valign="top">[a26]</td> <td valign="top"> A. Joyal, R. Street, "Braided tensor categories" ''Adv. Math.'' , '''102''' (1993) pp. 20–78</td></tr><tr><td valign="top">[a27]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a28]</td> <td valign="top"> M.M. Kapranov, V.A. Voevodsky, "Groupoids and homotopy types" ''Cah. Topol. Géom. Diff. Cat.'' , '''32''' (1991) pp. 29–46</td></tr><tr><td valign="top">[a29]</td> <td valign="top"> M.M. Kapranov, V.A. Voevodsky, "$2$-Categories and Zamolodchikov tetrahedra equations" , ''Proc. Symp. Pure Math.'' , '''56''' , Amer. Math. Soc. (1994) pp. 177–259</td></tr><tr><td valign="top">[a30]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a31]</td> <td valign="top"> C. Kassel, "Quantum groups" , ''Graduate Texts Math.'' : 155 , Springer (1995)</td></tr><tr><td valign="top">[a32]</td> <td valign="top"> G.M. Kelly, "Basic concepts of enriched category theory" , ''Lecture Notes London Math. Soc.'' : 64 , Cambridge Univ. Press (1982)</td></tr><tr><td valign="top">[a33]</td> <td valign="top"> V. Kharlamov, V. Turaev, "On the definition of the $2$-category of $2$-knots" ''Transl. Amer. Math. Soc.'' , '''174''' (1996) pp. 205–221</td></tr><tr><td valign="top">[a34]</td> <td valign="top"> L. Langford, "$2$-Tangles as a free braided monoidal $2$-category with duals" ''PhD Thesis Univ. California at Riverside'' (1997)</td></tr><tr><td valign="top">[a35]</td> <td valign="top"> J.-L. Loday, "Spaces with finitely many non-trivial homotopy groups" ''J. Pure Appl. Algebra'' , '''24''' (1982) pp. 179–202</td></tr><tr><td valign="top">[a36]</td> <td valign="top"> M. Mackay, "Spherical $2$-categories and $4$-manifold invariants" ''Adv. Math.'' , '''143''' (1999) pp. 288–348</td></tr><tr><td valign="top">[a37]</td> <td valign="top"> S. MacLane, "Possible programs for categorists" , ''Lecture Notes Math.'' , '''86''' , Springer (1969) pp. 123–131</td></tr><tr><td valign="top">[a38]</td> <td valign="top"> F. Marmolejo, "Distributive laws for pseudomonads" ''Theory Appl. Categ.'' , '''5''' (1999) pp. 91–147</td></tr><tr><td valign="top">[a39]</td> <td valign="top"> P. May, "The geometry of iterated loop spaces" , ''Lecture Notes Math.'' , '''271''' , Springer (1972)</td></tr><tr><td valign="top">[a40]</td> <td valign="top"> M. McIntyre, T. Trimble, "The geometry of Gray-categories" ''Adv. Math.'' (to appear)</td></tr><tr><td valign="top">[a41]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a42]</td> <td valign="top"> A.J. Power, "Why tricategories?" ''Inform. Comput.'' , '''120''' (1995) pp. 251–262</td></tr><tr><td valign="top">[a43]</td> <td valign="top"> N.Yu. Reshetikhin, V.G. Turaev, "Ribbon graphs and their invariants derived from quantum groups" ''Comm. Math. Phys.'' , '''127''' (1990) pp. 1–26</td></tr><tr><td valign="top">[a44]</td> <td valign="top"> J.E. Roberts, "Mathematical aspects of local cohomology" , ''Proc. Colloq. Operator Algebras and Their Application to Math. Physics, Marseille 1977'' , CNRS (1979)</td></tr><tr><td valign="top">[a45]</td> <td valign="top"> M.C. Shum, "Tortile tensor categories" ''J. Pure Appl. Algebra'' , '''93''' (1994) pp. 57–110 (PhD Thesis Macquarie Univ. Nov. 1989)</td></tr><tr><td valign="top">[a46]</td> <td valign="top"> R. Street, "The algebra of oriented simplexes" ''J. Pure Appl. Algebra'' , '''49''' (1987) pp. 283–335</td></tr><tr><td valign="top">[a47]</td> <td valign="top"> R. Street, "Parity complexes" ''Cah. Topol. Géom. Diff. Cat.'' , '''32''' (1991) pp. 315–343 (Corrigenda: 35 (1994) 359-361)</td></tr><tr><td valign="top">[a48]</td> <td valign="top"> R. Street, "Categorical structures" M. Hazewinkel (ed.) , ''Handbook of Algebra'' , '''I''' , Elsevier (1996) pp. 529–577</td></tr><tr><td valign="top">[a49]</td> <td valign="top"> R. Street, "Higher categories, strings, cubes and simplex equations" ''Appl. Categorical Struct.'' , '''3''' (1995) pp. 29–77 and 303</td></tr><tr><td valign="top">[a50]</td> <td valign="top"> 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)</td></tr><tr><td valign="top">[a51]</td> <td valign="top"> T. Trimble, "The definition of tetracategory" ''Handwritten diagrams,'' , '''August''' (1995)</td></tr><tr><td valign="top">[a52]</td> <td valign="top"> V.G. Turaev, "The Yang–Baxter equation and invariants of links" ''Invent. Math.'' , '''92''' (1988) pp. 527–553</td></tr><tr><td valign="top">[a53]</td> <td valign="top"> R.F.C. Walters, "Sheaves on sites as Cauchy-complete categories" ''J. Pure Appl. Algebra'' , '''24''' (1982) pp. 95–102</td></tr></table> |
Revision as of 16:46, 1 July 2020
$n$-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
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Higher-dimensional category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Higher-dimensional_category&oldid=19090