We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category , there is a self enriched multi-fusion category giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the e…
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48 results for “Symmetric Monoidal Categories”
Involutive Hopf monoids yield surface invariants.
problem Involutive Hopf monoids in symmetric monoidal categories.
method Construction of invariants via (co)equalizers and images.
result Categorical generalization of quantum double models.
We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C, W, z, z^*) consist…
Internalizes Turaev's construction for TQFTs using ribbon categories.
problem Constructing TQFTs from ribbon categories.
method Using a special kind of tangle and morphisms defined between tensorial products of the coend, an internal TQFT is constructed.
result An internal TQFT is a lift of Turaev's construction, equivalent to vector spaces when modular.
It is well known that the opposite F^{op} of the category F of finitely generated free groups is a Lawvere theory for groups, and also that F is a free symmetric monoidal category on a commutative Hopf monoid, or, in other words, a PROP for commutative Hopf algebras. In this paper, we give a direct, combinatorial proof…
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.
It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of …
Study shows algebraic structure in 2-dimensional CW-complex cobordisms.
problem Characterize cobordisms of 2-dimensional CW-complexes.
method Algebraic characterisation using Hopf algebras and symmetric monoidal categories.
result Category of cobordisms is equivalent to a freely generated Hopf algebra.
The paper studies posets from decompositions in symmetric monoidal categories.
problem Understanding posets from decompositions in symmetric monoidal categories.
method Defining decompositions and partial decompositions, complexes of frames, partial bases, and ordered versions.
result Unified approach to combinatorics and homotopy type of posets and complexes.
We define and study the category of symmetric -webs. This category is a combinatorial description of the category of all finite dimensional quantum -modules. Explicitly, we show that (the additive closure of) the symmetric -spider is (braided monoidally) equivalent to …
Cube category simplifies set modeling.
problem Modeling set operations efficiently.
method Introducing interval-preserving monotone functions between finite Boolean lattices.
result Cube category facilitates model structures equivalent to simplicial sets.
Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the categ…
Study of generalized Legendrian racks and their GL-structures.
problem Characterizing and classifying generalized Legendrian racks.
method Algebraic analysis, classification of racks and GL-racks, tensor products.
result Classification of several infinite families of GL-racks.
Extends six operations to sheaves in any symmetric monoidal category.
problem Extending six operations to a broader class of sheaves.
method Develops formalism for sheaves in any closed symmetric monoidal ∞-category, proving properties of locally contractible geometric morphisms and relating pullbacks and colimits.
result Establishes the six functor formalism for a wider range of sheaves, including those with values in any closed symmetric monoidal ∞-category.
We construct a categorification of the maximal commutative subalgebra of the type Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functo…
We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …
Open 2D TFTs extend to closed theories with circle value as Hochschild homology.
problem Extending open 2D TFTs to closed theories.
method Using symmetric monoidal ∞-categories and Hochschild homology.
result Open 2D TFTs admit initial open-closed extensions.
We define a category of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to which induces an equivalence of categories. On the other hand, we show that is universal among ribbon c…
New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
New geometric variant of factorization homology for conformally flat manifolds.
problem Defining invariants of conformally flat manifolds.
method Introducing a metric-dependent geometric variant of factorization homology.
result Left Kan extensions of conformally flat -disk algebras define invariants of conformally flat manifolds.
3D HQFTs constructed using graded monoidal categories.
problem Constructing 3D HQFTs with specific targets.
method Using spherical χ-fusion categories and the state sum method.
result 3D HQFTs constructed with target Bχ.
The paper discusses strictification and non-strictification of monoidal categories.
problem Transforming monoidal categories into strict or non-strict ones.
method Explicit models and 2-adjunctions for strictification and non-strictification.
result Strictification and non-strictification are part of a pair of free-forgetful 2-adjunctions.
It is proved that the category of simplicial complete bornological spaces over carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…
The paper constructs semistrict monoidal 2-categories from foam evaluations.
problem Creating examples of semistrict monoidal 2-categories.
method Using a closed foam evaluation formula as input, the paper rigorously constructs semistrict monoidal 2-categories.
result The constructed monoidal 2-categories are semistrict, have duals and adjoints, and carry a spatial duality structure.
We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use…
New monoids tied to symmetric group and Jones/Brauer monoids discovered.
problem Understanding monoids attached to symmetric group and related monoids.
method Introduced ramified monoids and tied-like monoids, provided presentations.
result Found new monoids that cannot be described as ramified monoids.
The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.
problem Exploring the algebraic structure of the conformal Laplacian in 2D.
method Using prefactorization algebras and Green functions.
result In 2D, the conformal Laplacian's algebraic structure is revealed through a central charge.
We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's funct…
Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is e…
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
A new category generates 1D tangle invariants.
problem Developing a new category for 1D tangles.
method Proving a new -category has universal mapping property.
result The new category generates link invariants.
Heaps are para-associative ternary operations bijectively exemplified by groups via the operation . They are also ternary self-distributive, and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories…
Monoidal categorifies genus zero skein algebra using K-theory.
problem Relating skein algebra to K-theory.
method Using quantized K-theoretic Coulomb branch and Grothendieck ring of equivariant coherent sheaves.
result Monoidal categorification of the skein algebra.
Theory for algebraic data on categories via concentration structures.
problem Defining algebraic structures on categories.
method Introducing concentration structures and concentration monoids.
result Every group can be represented as a concentration monoid of a trivial category.
We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in (arXiv:1202.3553) including the Kashaev invariant of links. Here the modular categor…
The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…
Constructs non-unital monoidal category of contact manifolds and Legendrian correspondence calculus
problem Constructing a non-unital monoidal category of contact manifolds without contact forms
method Developing contact topology without contact forms and defining the star product
result Proving the associativity of the star product and the pentagon axiom
We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived …
We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concept…
Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial…
Develops graphical calculus for monoidal categories with twisted pivotal structures.
problem Constructing modules for surfaces with Morse functions or foliations.
method Graphical calculus and string nets for monoidal categories with twisted pivotal structures.
result Twisted string net modules assemble in an oriented categorified 2-TQFT.
Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
DisCoPyro combines category theory with machine learning for program learning.
problem Applying category theory to machine learning tasks.
method Introducing DisCoPyro, a framework combining categorical structures with amortized variational inference.
result DisCoPyro can be applied in program learning for variational autoencoders and potentially contributes to AGI.
Generative model for morphisms in free categories learns from wiring diagrams.
problem Learning and generating morphisms in free monoidal categories.
method Probabilistic generative model using variational inference and maximum likelihood.
result The model achieves competitive performance on the Omniglot dataset.
Functor connects 4D 2-handlebodies to ribbon categories, detecting non-deformation diffeomorphisms.
problem Detecting non-deformation diffeomorphisms in 4D 2-handlebodies.
method Constructs a braided monoidal functor from 4D 2-handlebodies to unimodular ribbon categories.
result Functor detects non-deformation diffeomorphisms when is not semisimple and is not factorizable.
We study the algebraic structures of the virtual singular braid monoid, , and the virtual singular pure braid monoid, . The monoid is the splittable extension of by the symmetric group . We also construct a representation of .
It is shown that the multiplicative monoids of Temperley-Lieb algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the…
In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting cer…