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.

The notion of semi-abelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abelian-group and module theory. In modern terms, semi-abelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to "old" exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semi-abelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar non-abelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...

Moggi’s Computational Monads and Power et al’s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages such as exceptions, side-effects, input/output and continuations. We present generalisations of both constructs, which we call parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters. Examples of such are composable continuations, side-effects where the type of the state varies and input/output where the range of inputs and outputs varies. By also considering monoidal parameterisation, we extend the range of effects to cover separated side-effects and multiple independent streams of I/O. We also present two typed λ-calculi that soundly and completely model our categorical definitions — with and without monoidal parameterisation — and act as prototypical languages with parameterised effects.

Given a braided tensor ∗-category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗-category with conjugates and an irreducible unit. (A ∗-category is a category enriched over VectC with positive ∗-operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no non-trivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗-category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1

We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.

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 2-form electromagnetism to a kind of ‘higher-dimensional Yang–Mills theory’. It turns out that to do this, one should replace the Lie group by a ‘Lie 2-group’, 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 2-bundles ’ for any Lie 2-group C and do gauge theory in this new context. Here we only consider trivial 2-bundles, where a connection consists of a g-valued 1-form together with an h-valued 2-form, and its curvature consists of a g-valued 2-form together with a h-valued 3-form. 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 self-dual solutions in five dimensions. 1

We describe an interesting relation between Lie 2-algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras gk each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group having gk as its Lie 2-algebra, except when k = 0. Here, however, we construct for integral k an infinite-dimensional Lie 2-group PkG whose Lie 2-algebra is equivalent to gk. The objects of PkG are based paths in G, while the automorphisms of any object form the level-k Kac– Moody central extension of the loop group ΩG. This 2-group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group |PkG | that is an extension of G by K(Z, 2). When k = ±1, |PkG | can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), |PkG | is none other than String(n). 1 1

There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate "dualizing object". Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. We give a categorical discussion of such results. One essential point is to di#erentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely di#erent, framework that arises in algebraic topology.

We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalising the one introduced by Squier for word rewriting systems. We characterise this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs.

Abstract. We study Galois extensions M (co-)H ⊂ M for H-(co)module algebras M if H is a Frobenius Hopf algebroid. The relation between the action and coaction pictures is analogous to that found in Hopf-Galois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various classical theorems of Kreimer-Takeuchi, Doi-Takeuchi and Cohen-Fischman-Montgomery. We find that the Galois extensions N ⊂ M over some Frobenius Hopf algebroid are precisely the balanced depth 2 Frobenius extensions. We prove that the Yetter-Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by a slightly generalized Brzeziński-Militaru Theorem. Contravariant ”fiber functors ” are used to prove an analogue of Ulbrich’s Theorem and to get a monoidal embedding of the module category ME of the endomorphism Hopf algebroid E = End NMN into NM op N. 1.