Categorifying the Knizhnik–Zamolodchikov connection

In the context of higher gauge theory, we construct a flat and fake flat 2-connection, in the configuration space of n particles in the complex plane, categorifying the Knizhnik–Zamolodchikov connection. To this end, we define the differential crossed module of horizontal 2-chord diagrams, categorifying the Lie algebra of horizontal chord diagrams in a set of n parallel copies of the interval. This therefore yields a categorification of the 4-term relation. We carefully discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, which makes it possible to formulate the notion of an infinitesimal 2-R matrix in a differential crossed module.     

Crossed Modules for Lie 2-Algebras

We define the notion of crossed modules for Lie 2-algebras. To a given crossed module, we associate a strict Lie 3-algebra structure on its mapping cone complex and a strict Lie 2-algebra structure on its derivations. Finally, we classify strong crossed modules by means of the third cohomology group of Lie 2-algebras.     

Categorified quantum sl2 is an inverse limit of flag 2-categories

We prove that categorified quantum sl₂ is an inverse limit of Flag 2-categories defined using cohomology rings of iterated flag varieties. This inverse limit is an instance of a 2-limit in a bicategory giving rise to a universal property that characterizes the categorification of quantum sl₂ uniquely up to equivalence. As an application we characterize all bimodule homomorphisms in the Flag 2-category and prove that on the homological level the categorified quantum Casimir of sl₂ acts appropriately on these 2-representations.     

Fiat categorification of the symmetric inverse semigroup $$\textit{IS}_n$$ and the semigroup $$F^*_n$$Fn∗

Starting from the symmetric group \(S_n\), we construct two fiat 2-categories. One of them can be viewed as the fiat “extension” of the natural 2-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other 2-category can be viewed as the fiat “extension” of the 2-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.     

Limits for Lax Morphisms

We investigate limits in the 2-category of strict algebras and lax morphisms for a 2-monad. This includes both the 2-category of monoidal categories and monoidal functors as well as the 2-category of monoidal categories and opomonoidal functors, among many other examples.Mathematics Subject Classifications (2000) 18D05, 18C20, 18C15, 18A30.     

New Turaev Braided Group Categories and Group Schur–Weyl Duality

For a group G, we describe a new construction of a Turaev braided G-category with a particular braided monoidal subcategory and then we study the structure of a Hopf algebra in this subcategory. As an application, we establish a generalized G-Schur–Weyl duality between certain Turaev G-algebra and the symmetric group algebra.     

2-Groups, trialgebras and their Hopf categories of representations

A strict 2-group is a 2-category with one object in which all morphisms and all 2-morphisms have inverses. 2-Groups have been studied in the context of homotopy theory, higher gauge theory and Topological Quantum Field Theory (TQFT). In the present article, we develop the notions of trialgebra and cotrialgebra, generalizations of Hopf algebras with two multiplications and one comultiplication or vice versa, and the notion of Hopf categories, generalizations of monoidal categories with an additional functorial comultiplication. We show that each strict 2-group has a ‘group algebra’ which is a cocommutative trialgebra, and that each strict finite 2-group has a ‘function algebra’ which is a commutative cotrialgebra. Each such commutative cotrialgebra gives rise to a symmetric Hopf category of corepresentations. In the semisimple case, this Hopf category is a 2-vector space according to Kapranov and Voevodsky. We also show that strict compact topological 2-groups are characterized by their C^*-cotrialgebras of ‘complex-valued functions’, generalizing the Gel'fand representation, and that commutative cotrialgebras are characterized by their symmetric Hopf categories of corepresentations, generalizing Tannaka-Kre?ˇn reconstruction. Technically, all these results are obtained using ideas from functorial semantics, by studying models of the essentially algebraic theory of categories in various base categories of familiar algebraic structures and the functors that describe the relationships between them.     

The formal theory of monoidal monads

We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg–Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. We explain at the end of the paper that a similar phenomenon occurs in many other situations.     

On the cobordism and commutative monoid with cancellation approaches to conformal field theory

In the late 1980s, Graeme Segal axiomatized conformal field theory in terms of a cobordism category. In that same preprint he outlined a more symmetric trace approach, which was recently rigorized in terms of pseudo algebras over a 2-theory. In this paper, we treat the cobordism approach in the pseudo algebra context. We introduce a new algebraic structure on a bicategory, called a pseudo 2-algebra over a theory, as a means of comparison for the two approaches. The main result states that the 2-category of pseudo algebras over a fixed 2-theory is biequivalent to the 2-category of pseudo 2-algebras over a fixed theory in certain situations.     

A new duality theory for compact groups

Summary The Tannaka-Krein duality theory characterizes the category (G) of finite-dimensional, continuous, unitary representations of a compact group as a subcategory of the category of Hilbert spaces. We prove a more powerful result characterizing (G) as an abstract category: every strict symmetric monoidalC ^*-category with conjugates which has subobjects and direct sums and for which theC ^*-algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category (G) for a compact groupG unique up to isomorphism.Research supported by the Ministero della Pubblica Istruzione and CNR-GNAFA     

Not every pseudoalgebra is equivalent to a strict one

We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical and higher-categorical coherence theory. In particular, any nontrivially braided monoidal category gives an example of a pseudoalgebra that is not equivalent to a strict one.     

Accessible aspects of 2-category theory

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the structure of monoidal, but not strict monoidal, categories) then the 2-category in question is accessible. Furthermore, we explore the flexible limits that such 2-categories possess and their interaction with filtered colimits.     

Infinite loop spaces,and coherence for symmetric monoidal bicategories

This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one.     

Crossed Products in Weak Contexts

We define the general notion of crossed products in a weak context, which generalizes the ones defined by Blattner, Cohen and Montgomery, Doi and Takeuchi in the context of Hopf algebras and the one given by Brzeziński. Also, the crossed products obtained by the authors, for weak Hopf algebras living in a symmetric monoidal category and weak C-cleft extensions associated to weak entwined structures, are particular instances of this theory.     

Cleft extensions in braided categories

Majid in [14] and Bespalov in [2] obtain a braided interpretation of Radford’s theorem about Hopf algebras with projection ([19]). In this paper we introduce the notion of H-cleft comodule (module) algebras (coalgebras) for a Hopf algebra H in a braided monoidal category, and we characterize it as crossed products (coproducts). This allows us give very short proofs for know results in our context, and to introduce others stated for the category of R-modules about of Hopf algebra extensions. In particular we give a proof of the result by Bespalov [2] for a braided monoidal category with co(equalizers).     

Bispectral quantum Knizhnik�CZamolodchikov equations for arbitrary root systems

The bispectral quantum Knizhnik–Zamolodchikov (BqKZ) equation corresponding to the affine Hecke algebra H of type A _N-1 is a consistent system of q-difference equations which in some sense contains two families of Cherednik’s quantum affine Knizhnik–Zamolodchikov equations for meromorphic functions with values in principal series representations of H. In this paper, we extend this construction of BqKZ to the case where H is the affine Hecke algebra associated with an arbitrary irreducible reduced root system. We construct explicit solutions of BqKZ and describe its correspondence to a bispectral problem involving Macdonald’s q-difference operators.     

Monoidal infinity category of complexes from Tannakian viewpoint

In this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable $\infty $ -categories of quasi-coherent complexes. It can be viewed as a derived analogue of Tannaka duality. As a consequence, we deduce that an algebraic stack satisfying a certain condition can be recovered from the symmetric monoidal stable $\infty $ -category of quasi-coherent complexes with tensor operation.     

Higher-dimensional algebra IV: 2-tangles

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 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.     

Duality for Knizhnik–Zamolodchikov and Dynamical Equations

We consider the Knizhnik–Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of (gl _k ,gl _n ) duality. We show that the KZ and dynamical equations naturally exchange under the duality.