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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 A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor.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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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor.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
Lie theory for nilpotent L∞algebras
 Ann. Math
"... Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric n ..."
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Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric nsimplex ∆ n. In [20], Sullivan reformulated Quillen’s
Convex hull realizations of the multiplihedra
, 2007
"... Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents ..."
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Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents
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).
Enrichment as Categorical Delooping I: Enrichment Over Iterated Monoidal Categories
, 2008
"... Joyal and Street note in their paper on braided monoidal categories [10] that the 2–category V–Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. What is meant by “based upon ” here will be made more clear in the prese ..."
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Cited by 4 (4 self)
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Joyal and Street note in their paper on braided monoidal categories [10] that the 2–category V–Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. What is meant by “based upon ” here will be made more clear in the present paper. The exception that they mention is the case in which V is symmetric, which leads to V–Cat being symmetric as well. The symmetry in V–Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V–Cat are in turn related to a categorical analogue of topological delooping first mentioned by Baez and Dolan in [1]. To do so I need to pass to a more general setting than braided and symmetric categories – in fact the k–fold monoidal categories of Balteanu et al in [3]. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k–fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k–fold monoidal category, V–Cat becomes a (k − 1)–fold monoidal 2– category in a canonical way. I indicate how this process may be iterated by enriching over V–Cat, along the way defining the 3–category of categories enriched over V–Cat. In the next paper I hope to make precise the n– dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V–Cat.
Crossed module bundle gerbes; classification, string group and differential geometry. Available as arXiv:math/0510078v2
"... We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module (H → D) there is a simplicial group NC(H→D), the nerve of the 1category defined by the crossed module and its geometric realization NC(H→D). Equivalence classes of princi ..."
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We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module (H → D) there is a simplicial group NC(H→D), the nerve of the 1category defined by the crossed module and its geometric realization NC(H→D). Equivalence classes of principal bundles with structure group NC(H→D)  are shown to be onetoone with stable equivalence classes of what we call crossed module bundle gerbes. We can also associate to a crossed module a 2category ˜ C(H→D). Then there are two equivalent ways how to view classifying spaces of NC(H→D)bundles and hence of NC(H→D)bundles and crossed module bundle gerbes. We can either apply the Wconstruction to NC(H→D) or take the nerve of the 2category ˜ C(H→D). We discuss the string group and string structures from this point of view. Also a simplicial principal bundle can be equipped with a simplicial connection and a Bfield. It is shown how in the case of a simplicial principal NC(H→D)bundle these simplicial objects give the bundle gerbe connection and the bundle gerbe Bfield.
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
A full and faithful nerve for 2categories
 Appl. Categ. Structures
, 2005
"... We prove that there is a full and faithful nerve functor defined on the category 2Catlax of 2categories and (normal) lax 2functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the socalled geometric nerve of a 2category or of a 2groupoid. We al ..."
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We prove that there is a full and faithful nerve functor defined on the category 2Catlax of 2categories and (normal) lax 2functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the socalled geometric nerve of a 2category or of a 2groupoid. We also show that (normal) lax 2natural transformations produce homotopies of a special kind, and that two lax 2functors from a 2category to a 2groupoid have homotopic nerves if and only if there is a lax 2natural transformation between them. 1
2DENDROIDAL SETS
"... that the objects of Ω are trees and a morphism t → t ′ is an operad map from Ω(t) to Ω(t ′), that is, from the operad generated by the vertices of t to that generated by the vertices of t ′. [Globular pasting diagrams are pasting diagrams which correspond to trees with height. Note that the two sort ..."
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that the objects of Ω are trees and a morphism t → t ′ is an operad map from Ω(t) to Ω(t ′), that is, from the operad generated by the vertices of t to that generated by the vertices of t ′. [Globular pasting diagrams are pasting diagrams which correspond to trees with height. Note that the two sorts of trees just mentioned are not directly related. For one thing there are two sorts of tree composition around, one the ordinary grafting, and the other the special composition that reflects composition of pasting diagrams. We won’t draw the trees with height unless they make definitions or proofs more efficient. Sources are Batanin [4] and Leinster [13]. For examples of pasting diagrams together with their trees see page 8 of [6] at
ENRICHED 2NATURAL TRANSFORMATIONS, MODIFICATIONS, AND HIGHER MORPHISMS
"... Abstract. We review enriched 2natural transformations, modifications, and higher morphisms in the context of a symmetric monoidal ncategory V. The goal is to discern what sort of algebraic structure these actually comprise. ..."
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Abstract. We review enriched 2natural transformations, modifications, and higher morphisms in the context of a symmetric monoidal ncategory V. The goal is to discern what sort of algebraic structure these actually comprise.