In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functo…
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16 results for “2-functor”
Good atlases are defined for effective orbifolds, and a spark complex is constructed on each good atlas. It is proved that this process is 2-functorial with compatible systems playing as morphisms between good atlases, and that the spark character 2-functor factors through this 2-functor.
We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links.
We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in se…
Defines a new field theory in 1+1+1 dimensions.
problem Developing a new field theory in 1+1+1 dimensions.
method Defines an extended field theory as a quasi 2-functor with values in a completed 2-category, .
result Extends existing theories and provides a real analog of a construction by Moore and Tachikawa.
Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid and we prove that t…
A modular functor is constructed from non-semisimple 3d TFTs.
problem Constructing modular functors from non-semisimple 3d topological field theories.
method Using a 3d TFT defined in [arXiv:1912.02063], a symmetric monoidal 2-functor is constructed from a 2-category of bordisms to a 2-category of finite linear categories.
result A modular functor is explicitly described as a symmetric monoidal 2-functor.
A model structure is defined on the category of derived differentiable schemes, and it is used to analyse the truncation 2-functor from derived manifolds to d-manifolds. It is proved that the induced 1-functor between the homotopy categories is full and essentially surjective, giving a bijection between the sets of equ…
Defines parallel 2-transport and 2-group bundles, proving new theorems.
problem Understanding parallel transport in higher dimensions.
method Introduces a new 2-category of 2-group torsors and defines parallel 2-transport as a 2-functor.
result Proves non-Abelian Stokes and Ambrose-Singer theorems for 2-transport.
We study a variation of Turaev's homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space and define an $\cS_X$-structure to be a monoidal 2-functor from this to the 2-category of idempotent-complete additive -linear categories. We initiate the study of…
We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrat…
This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the und…
We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms al…
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…
Third paper in series defines monopole Floer homology and gluing theorem for 3-manifolds.
problem Define and prove gluing theorem for monopole Floer homology.
method Define monopole Floer homology, establish gluing theorem for disconnected boundary.
result Floer homology detects Thurston norm and fiberness of 3-manifolds.
In this paper, we 'construct' a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered category associated to…