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''$n$-category''
 
''$n$-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 $0 \leq k &lt; m \leq n$, 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 $0 \leq h &lt; k &lt; m \leq n$, 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 ($k &lt; m \leq n$) 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 $0 &lt; m \leq n$, 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).
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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 $n$-'''Cat''', 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]]].
 
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 ($n &gt; 2$) 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]].
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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.
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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 $2$-'''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 [[#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 '''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 [[#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 [[operad]]s 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]]].
 
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 [[operad]]s 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]]].
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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 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 $n$-categories ($n &gt; 2$) 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]]].
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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 $n$-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.
 
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====
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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>
+
<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>

Latest revision as of 00:41, 15 February 2024

$n$-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

[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
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How to Cite This Entry:
Higher-dimensional category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Higher-dimensional_category&oldid=52653
This article was adapted from an original article by Ross Street (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article