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24
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 46 (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.
Higherdimensional algebra IV: 2Tangles
"... Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we p ..."
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Cited by 34 (10 self)
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Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2category is the ‘free semistrict braided monoidal 2category with duals on one unframed selfdual object’. By this universal property, any unframed selfdual object in a braided monoidal 2category with duals determines an invariant of 2tangles in 4 dimensions. 1
The EckmannHilton argument, higher operads and Enspaces, available at http://www.ics.mq.edu.au
 mbatanin/papers.html of Homotopy and Related Structures
"... The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of ..."
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Cited by 33 (5 self)
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The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2category, then its Homset is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an noperad A in the author’s sense there exists a symmetric operad S n (A) called the nfold suspension of A such that the
Finite groups, spherical 2categories, and 4manifold invariants. arXiv:math.QA/9903003
"... In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], althou ..."
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Cited by 16 (5 self)
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In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the statesum invariants of Birmingham and Rakowski [11, 12, 13], who studied DijkgraafWitten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3types, such as [15], for example. 1 1
A Tensor Product for GrayCategories
, 1999
"... In this paper I extend Gray's tensor product of 2categories to a new tensor product of Graycategories. I give a description in terms of generators and relations, one of the relations being an "interchange" relation, and a description similar to Gray's description of his tensor ..."
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Cited by 6 (2 self)
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In this paper I extend Gray's tensor product of 2categories to a new tensor product of Graycategories. I give a description in terms of generators and relations, one of the relations being an "interchange" relation, and a description similar to Gray's description of his tensor product of 2categories. I show that this tensor product of Graycategories satisfies a universal property with respect to quasifunctors of two variables, which are defined in terms of laxnatural transformations between Graycategories. The main result is that this tensor product is part of a monoidal structure on GrayCat, the proof requiring interchange in an essential way. However, this does not give a monoidal (bi)closed structure, precisely because of interchange. And although I define composition of laxnatural transformations, this composite need not be a laxnatural transformation again, making GrayCat only a partial (GrayCat)\Omega  CATegory.
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
The EckmanHilton argument and higher operads, preprint
, 2006
"... To the memory of my father. The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy ..."
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Cited by 3 (2 self)
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To the memory of my father. The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2category, then its Homset is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the EckmanHilton type argument in terms of the calculation of left Kan extensions in an appropriate 2category. Then we apply this scheme to the case of noperads in the author’s sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of noperads to the
Balanced Coalgebroids
, 2000
"... A balanced coalgebroid is a V op category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of ..."
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Cited by 2 (0 self)
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A balanced coalgebroid is a V op category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of representations. This includes the reconstruction of dual quasibialgebras, quasitriangular dual quasibialgebras, and balanced quasitriangular dual quasibialgebras; the latter of which is a quantum group when equipped with a compatible antipode. As an application we construct a balanced coalgebroid whose category of representations is equivalent to the symmetric monoidal category of chain complexes. The appendix provides the definitions of a braided monoidal bicategory and sylleptic monoidal bicategory.