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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
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Cited by 141 (14 self)
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For a copy with the handdrawn figures please email
Higherdimensional algebra VI: Lie 2algebras,
, 2004
"... The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We ..."
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Cited by 44 (12 self)
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The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We define a ‘semistrict Lie 2algebra ’ to be a 2vector space L equipped with a skewsymmetric bilinear functor [·, ·]: L × L → L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the ‘Jacobiator’, which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang–Baxter equation. We construct a 2category of semistrict Lie 2algebras and prove that it is 2equivalent to the 2category of 2term L∞algebras in the sense of Stasheff. We also study strict and skeletal Lie 2algebras, obtaining the former from strict Lie 2groups and using the latter to classify Lie 2algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finitedimensional Lie algebra g a canonical 1parameter family of Lie 2algebras g � which reduces to g at � = 0. These are closely related to the 2groups G � constructed in a companion paper.
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 26 (2 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Higher YangMills theory
"... Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2form electromagnetism to a kind of ‘higherdimensional Yang–Mills theory’. It turns out that to do this, one should repla ..."
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Cited by 20 (1 self)
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Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2form electromagnetism to a kind of ‘higherdimensional Yang–Mills theory’. It turns out that to do this, one should replace the Lie group by a ‘Lie 2group’, which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a ‘Lie crossed module’: a pair of Lie groups G, H with a homomorphism t: H → G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing’s ideas on the geometry of nonabelian gerbes, one can define ‘principal 2bundles ’ for any Lie 2group C and do gauge theory in this new context. Here we only consider trivial 2bundles, where a connection consists of a gvalued 1form together with an hvalued 2form, and its curvature consists of a gvalued 2form together with a hvalued 3form. We generalize the Yang–Mills action for this sort of connection, and use this to derive ‘higher Yang– Mills equations’. Finally, we show that in certain cases these equations admit selfdual solutions in five dimensions. 1
Measurable categories and 2groups
 Appl. Cat. Struct
"... Abstract: Using the theory of measurable categories developped in [Yet03], we provide a notion of representations of 2groups more wellsuited to physically and geometrically interesting examples than that using 2VECT (cf. [KV94]). Using this theory we sketch a 2categorical approach to the states ..."
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Cited by 12 (0 self)
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Abstract: Using the theory of measurable categories developped in [Yet03], we provide a notion of representations of 2groups more wellsuited to physically and geometrically interesting examples than that using 2VECT (cf. [KV94]). Using this theory we sketch a 2categorical approach to the statesum model for Lorentzian quantum gravity proposed in [CY03], and suggest stateintegral constructions for 4manifold invariants. 1
Representation theory of 2groups on Kapranov and Voevodsky’s 2vector spaces
 Adv. Math
"... In this paper the 2category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2group G on (some version of) Kapranov and Voevodsky’s 2category of (complex) 2vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invaria ..."
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Cited by 10 (1 self)
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In this paper the 2category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2group G on (some version of) Kapranov and Voevodsky’s 2category of (complex) 2vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants π0(G), π1(G) and [α]∈H 3 (π0(G), π1(G)) classifying G. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain that the monoidal category
The Classifying Space of a Topological 2Group
, 2008
"... Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal G ..."
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Cited by 4 (1 self)
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Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal Gbundles over M are classified by either the Čech cohomology ˇ H 1 (M, G) or the set of homotopy classes [M, BG], where BG is the classifying space of G. Here we review work by Bartels, Jurčo, Baas–Bökstedt–Kro, and others generalizing this result to topological 2groups and even topological 2categories. We explain various viewpoints on topological 2groups and the Čech cohomology ˇ H 1 (M, G) with coefficients in a topological 2group G, also known as ‘nonabelian cohomology’. Then we give an elementary proof that under mild conditions on M and G there is a bijection ˇH 1 (M, G) ∼ = [M, BG] where BG  is the classifying space of the geometric realization of the nerve of G. Applying this result to the ‘string 2group ’ String(G) of a simplyconnected compact simple Lie group G, it follows that principal String(G)2bundles have rational characteristic classes coming from elements of H ∗ (BG, Q)/〈c〉, where c is any generator of H 4 (BG, Q).
Lie 2algebras
, 2004
"... I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times ..."
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Cited by 4 (1 self)
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I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times during my graduate studies. I also thank my ‘mathematical brothers’: Miguel CarriónÁlvarez, Toby Bartels, Jeffrey Morton, and Derek Wise for their friendship and engaging, educational conversations. I am grateful for the assistance of Aaron Lauda in drawing various braid diagrams, and thank Ronnie Brown, Andrée Ehresmann, Thomas Larsson, James Stasheff, J. Scott Carter, and Masahico Saito for helpful discussions and correspondence. Finally, I am extremely appreciative of the love and support of my family, friends, and former professors during my time as a graduate student. I certainly could not have accomplished all that I have without them. iii ABSTRACT OF THE DISSERTATION
Representation theory of 2groups on finite dimensional 2vector spaces, in preparation
, 2004
"... In this paper we unfold the 2category structure of the representations of a (strict) 2group on (a suitable version of) Kapranov and Voevodsky’s 2category of finite dimensional 2vector spaces and we discuss the relationship with classical representation theory of groups on finite dimensional vect ..."
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Cited by 4 (1 self)
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In this paper we unfold the 2category structure of the representations of a (strict) 2group on (a suitable version of) Kapranov and Voevodsky’s 2category of finite dimensional 2vector spaces and we discuss the relationship with classical representation theory of groups on finite dimensional vector spaces. In particular, we prove that the monoidal category of representations of any group G appears as a full subcategory of the category of endomorphisms of a particular object in the 2category of representations of G when G is thought of as a 2group with only identity arrows. As an easy consequence of the unfolding process, we also see that every 2group with a compact Lie group as base group has a rank one representation faithful with respect to the base group, contrary to a claim by Barrett and Mackaay (unpublished work). 1
Categorification
 Contemporary Mathematics 230. American Mathematical Society
, 1997
"... Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘c ..."
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Cited by 4 (1 self)
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Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘coherence laws’. Iterating this process requires a theory of ‘ncategories’, algebraic structures having objects, morphisms between objects, 2morphisms between morphisms and so on up to nmorphisms. After a brief introduction to ncategories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle ncategories, cobordism ncategories, and the homotopy ntypes of the loop spaces Ω k S k. We conclude by describing a definition of weak ncategories based on the theory of operads. 1