The theory of Lie algebras can be categorified starting from a new notion of ‘2-vector space’, which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2-morphisms. We define a ‘semistrict Lie 2-algebra ’ to be a 2-vector space L equipped with a skew-symmetric 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 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang–Baxter equation. We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L∞-algebras in the sense of Stasheff. We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finite-dimensional Lie algebra g a canonical 1-parameter family of Lie 2-algebras g � which reduces to g at � = 0. These are closely related to the 2-groups G � constructed in a companion paper.

We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...

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 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the ‘free semistrict braided monoidal 2-category with duals on one unframed self-dual object’. By this universal property, any unframed self-dual object in a braided monoidal 2-category with duals determines an invariant of 2-tangles in 4 dimensions. 1

Recent developments in higher-dimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2-categories, and a center construction for monoidal 2-categories which gives a braided monoidal 2-category. I give generalized center constructions for braided and sylleptic monoidal 2-categories which give sylleptic and symmetric monoidal 2-categories respectively, and I correct some errors in the original center construction for monoidal 2-categories. 1 Introduction The initial motivation for the study of braided monoidal categories was twofold: from homotopy theory, where braided monoidal categories of a particular kind arise as algebraic 3-types of arc-connected, simply connected spaces, and from higher-dimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16]. These motivations have subsequently been brought together by the definition of tricategori...

In this paper the concurrent semantics of double-pushout (DPO) graph rewriting, which is classically defined in terms of shift-equivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bi-category. In contrast to a previous attempt based on 2-categories, the use of bi-categories allows to define rewriting on concrete graphs. Thus, the problem of composition of isomorphism classes of rewriting sequences is avoided. Moreover, as a first step towards the recovery of the full expressive power of the formalism via a purely algebraic description, the concept of disconnected rules is introduced, i.e., rules whose interface graphs are made of disconnected nodes and edges only. It is proved that, under reasonable assumptions, rewriting via disconnected rules enjoys similar concurrency properties like in the classical approach.

. We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd a m a dD. We give the definition in any Gray-category. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZ-doctrine. 1. Introduction Free co-completions of categories under suitable classes of colimits were the motivating examples for the definition of KZ-doctrines. We introduce in this paper a not-strict version of such doctrines defined via a fully faithful adjoint string. Thus, a non-strict KZ-doctrine on a 2-category K consists of a normal endo homomorphism D : K \Gamma! K, and strong transformations d : 1K \Gamma! D, and m : DD \Gamma! D in such a way that Dd a m a dD forms a fully faithful adjoint string, satisfying one equation involving the unit of...

Abstract A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term "quantum category"in a braided monoidal category with equalizers distributed over by tensoring with an object. The definition of antipode for a bialgebroid is less resolved in the literature. Our suggestion is that the kind of dualization occurring in Barr's starautonomous categories is more suitable than autonomy ( = compactness = rigidity). This leads to our definition of quantum groupoid intended as a "Hopf algebra with several objects". 1.

Given a 2-category K admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category L with a 2-monad S on it such that: • S has the adjoint-pseudo-algebra property. • The 2-categories of pseudo-algebras of S and T are equivalent. Thus, coherent structures (pseudo-T-algebras) are transformed into universally characterised ones (adjoint-pseudo-S-algebras). The 2-category L consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudo-S-algebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudo-algebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudo-functors into Cat.

Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.

. We define distributive laws between pseudomonads in a Gray-category A, as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the Gray-category PSM(A) of pseudomonads in A, and define a lifting to be a pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure with respect to two given pseudomonads. We show how to obtain a pseudomonad with compatible structure from a distributive law, how to get a lifting from a pseudomonad with compatible structure, and how to obtain a distributive law from a lifting. We show that one triangle suffices to define a distributive law in case that one of the pseudomonads is a (co-)KZ-doctrine and the other a KZ-doctrine. 1. Introduction Distributive laws for monads were introduced by J. Beck in [2]. As pointed out by G. M. Kelly in [7], strict distributive laws for higher dimensional monads are rare. We need then a study ...