How strict is strictification?

The subject of this paper is the higher structure of the strictification adjunction, which relates the two fundamental bases of three-dimensional category theory: the Gray-category of 2-categories and the tricategory of bicategories. We show that – far from requiring the full weakness provided by the definitions of tricategory theory – this adjunction can be strictly enriched over the symmetric closed multicategory of bicategories defined by Verity. Moreover, we show that this adjunction underlies an adjunction of bicategory-enriched symmetric multicategories. An appendix introduces the symmetric closed multicategory of pseudo double categories, into which Verity's symmetric multicategory of bicategories embeds fully.     

CONCRETE CATEGORIES FOR WHICH FIBRE COMPLETIONS INDUCE TOPOLOGICAL COMPLETIONS

Abstract

Every topological category over an arbitrary base category X may be considered as a category of T-models with respect to some theory (i.e., functor) T from X into a category of complete lattices. Using this model-theoretic correspondence as our basic tool, we study initial and final completions of (co)fibration complete categories. For an arbitrary concrete category (A, U) over X, the process of order-theoretically completing each fibre does not usually yield an initial/final completion of (A, U). It is shown in this paper that for concrete categories which are assumed to be fibration and/or cofibration complete, initial and final completions can be constructed by completing the fibres. These completions are further shown to exhibit some interesting external properties.     

Formal Hopf algebra theory I: Hopf modules for pseudomonoids

In this article we develop some of the basic constructions of the theory of Hopf algebras in the context of autonomous pseudomonoids in monoidal bicategories. We concentrate on the notion of Hopf modules. We study the existence and the internalisation of this notion, called the Hopf module construction. Our main result is the equivalence between the existence of a left dualization for A (i.e., A is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case a Hopf module construction for A always exists. We recover from the general theory developed here results on coquasi-Hopf algebras.     

The bicategories of corings

To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projective on one side, together with a map of corings over the same base ring, lead to the notion of a module-morphism, which extends the notion of a morphism of corings (over different base rings). A module-morphism of corings induces functors between the categories of comodules. These functors are termed pull-back and push-out functors, respectively, and thus relate categories of comodules of different corings. We study when the pull-back functor is fully faithful and when it is an equivalence. A generalised descent associated to a morphism of corings is introduced. We define a category of module-morphisms, and show that push-out functors are naturally isomorphic to each other if and only if the corresponding module-morphisms are mutually isomorphic. All these topics are studied within a unifying language of bicategories and the extensive use is made of interpretation of corings as comonads in the bicategory Bim of bimodules and module-morphisms as 1-cells in the associated bicategories of comonads in Bim.     

Bounded G-theory with fibred control

We use filtered modules over a Noetherian ring and fibred bounded control on homomorphisms to construct a new kind of controlled algebra with applications in geometric topology. The resulting theory can be thought of as a “pushout” of bounded K-theory with fibred control and bounded G-theory constructed and used by the authors. Bounded G-theory was geared toward constructing a G-theoretic version of assembly maps and proving the Novikov injectivity conjecture for them. The G-theory with fibred control is needed in the study of surjectivity of the assembly map. The relation between the K- and G-theories is the classical one: K-theory is meaningful, however G-theory is easier to compute, and the relationship is expressed via a Cartan map. This map turns out to be an equivalence under very mild constraints in terms of metric geometry such as finite decomposition complexity. The fibred theory is certainly more complicated than the absolute theory. This paper contains the non-equivariant theory including fibred controlled excision theorems known to be crucial for computations.     

Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds

We apply the methods of C a?ld?raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds which are fibred by Lagrangian abelian surfaces. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain Lagrangian fibration and the derived category of twisted sheaves on its ‘mirror’ partner. As a corollary, we extend the original Fourier-Mukai transform to degenerations of abelian surfaces. Another consequence of the general theory is that the holomorphic symplectic four-fold and its mirror are connected by a one-parameter family of deformations through Lagrangian fibrations.     

Hodge cohomology of some foliated boundary and foliated cusp metrics

For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of L² harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of the foliation) is a Witt space and if the metric considered is a foliated cusp metric, then no such resolution is required.     

Measurable Categories

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [3] and [4] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are established along the way: first it is shown that all invertible additive bounded functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable transformations between the underlying Borel spaces and second we establish the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to σ-finite measures. The paper concludes with a general definition of measurable bicategories.     

Taut foliations in punctured surface bundles, I

Given a fibred, compact, orientable 3-manifold with single boundarycomponent, we show that a fibration with fiber surface of negativeEuler characteristic can be perturbed to yield taut foliationsrealizing an open interval of boundary slopes about the boundaryslope of the fibration. These taut foliations extend to tautfoliations in the corresponding surgery manifolds. 2000 MathematicsSubject Classification: primary 57M25; secondary 57R30.     

Point-Reflection Geometries, Geometric K-Loops and Unitary Geometries

There are reflection geometries to which correspond K- loops (P,+,F) with an incidence fibration F (= incidence partition), where F consists of proper subloops which are not groups. Such examples are derived from certain unitary geometries.     

Restrictions on symplectic fibrations

Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E₂. In the appendix, we prove nontriviality of certain Gromov-Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of  along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.     

A folk model structure on omega-cat

The primary aim of this work is an intrinsic homotopy theory of strict ω-categories. We establish a model structure on ωCat, the category of strict ω-categories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All objects are fibrant while free objects are cofibrant. We further exhibit model structures of this type on n-categories for arbitrary n∈N, as specializations of the ω-categorical one along right adjoints. In particular, known cases for n=1 and n=2 nicely fit into the scheme.     

Fibrations on four-folds with trivial canonical bundles

Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and prove that there is no such fibration in the sixth class. We classify all such fibrations whose generic fibre is the Jacobian of a genus two curve.     

Classifying spaces for braided monoidal categories and lax diagrams of bicategories

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well-understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves.     

Protomodular aspect of the dual of a topos

The structure of the dual of a topos is investigated under its aspect of a Barr exact and protomodular category. In particular the normal monomorphisms in the fibres of the fibration of pointed objects are characterized, and the change of base functors with respect to this same fibration are shown to reflect those normal monomorphisms.     

Universal properties of bicategories of polynomials

We establish the universal properties of the bicategory of polynomials, considering both cartesian and general morphisms between these polynomials. A direct proof of these universal properties would be impractical due to the complicated coherence conditions arising from polynomial composition; however, in this paper we avoid most of these coherence conditions using the properties of generic bicategories.In addition, we give a new proof of the universal properties of the bicategory of spans, and also establish the universal properties of the bicategory of spans with invertible 2-cells; showing how these properties may be used to describe the universal properties of polynomials.     

Symmetrisation of n-operads and compactification of real configuration spaces

It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym₁ given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author's theory of higher operads, the nonsymmetric operads are 1-operads and Sym₁ is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from n-operads to symmetric operads with the property that the category of one object, one arrow, …, one (n−1)-arrow algebras of an n-operad A is isomorphic to the category of algebras of Symn(A).In this paper we consider some geometrical and homotopical aspects of the symmetrisation of n-operads. We follow Getzler and Jones and consider their decomposition of the Fulton-Macpherson operad of compactified real configuration spaces. We construct an n-operadic counterpart of this compactification which we call the Getzler-Jones operad. We study the properties of Getzler-Jones operad and find that it is contractible and cofibrant in an appropriate model category. The symmetrisation of the Getzler-Jones operad turns out to be exactly the operad of Fulton and Macpherson. These results should be considered as an extension of Stasheff's theory of 1-fold loop spaces to n-fold loop spaces n?2. We also show that a space X with an action of a contractible n-operad has a natural structure of an algebra over an operad weakly equivalent to the little n-disks operad. A similar result holds for chain operads. These results generalise the classical Eckman-Hilton argument to arbitrary dimension.Finally, we apply the techniques to the Swiss-Cheese type operads introduced by Voronov and prove analogous results in this case.     

2-Cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors

In this paper we take up again the deformation theory for K-linear pseudofunctors initiated in Elgueta (Adv. Math. 182 (2004) 204-277). We start by introducing a notion of 2-cosemisimplicial object in an arbitrary 2-category and analyzing the corresponding coherence question, where the permutohedra make their appearance. We then describe a general method to obtain usual cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in the 2-category of small K-linear categories and prove that the deformation complex introduced in Elgueta (to appear) can be obtained by this method from a 2-cosemisimplicial object that can be associated to . Finally, using this 2-cosemisimplicial object of and a generalization to the context of K-linear categories of the deviation calculus introduced by Markl and Stasheff for K-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to the integrability of an nth-order deformation of indeed correspond to cocycles in the third cohomology group , a question which remained open in Elgueta (Adv. Math. 182 (2004) 204-277).