Clarifies sign ambiguity in Khovanov homology functoriality.
problem Sign ambiguity in Khovanov homology functoriality.
method Blanchet's oriented model and cobordisms with singularities.
result Strict functoriality established for the oriented model.
New formulas for knot invariants under specific surgeries.
problem Defining and proving splitting formulas for knot invariants.
method Defined a functorial extension of the Kricker invariant and proved splitting formulas for null Lagrangian-preserving surgery.
result Splitting formulas for the Kricker invariant under null Lagrangian-preserving surgery.
Extends Khovanov homology to surfaces with singularities.
problem Computing invariants for surfaces with singularities.
method Functorial extension of Khovanov homology to surfaces with double points.
result Induces a map between Khovanov homology groups of boundary links.
Khovanov spectra are shown to be functorial under certain conditions.
problem Understanding functoriality of Khovanov spectra.
method Proving functoriality up to homotopy and sign for Khovanov spectra.
result Khovanov spectra are functorial under specific conditions.
Functoriality proved for colored link invariants.
problem Link and tangle invariants functoriality proof.
method Functoriality proved for colored Khovanov-Rozansky invariants.
result Functoriality of colored link homologies proved.
Proves functoriality in geometric quantization for non-compact spin-c manifolds.
problem Non-compact spin-c manifolds in geometric quantization.
method Formal geometric quantization procedure for spin-c manifolds.
result Functoriality proven in spin-c setting.
Extends Khovanov bracket to link cobordisms, proving functoriality up to scalars.
problem Proving functoriality of Khovanov homology under link cobordisms.
method Extending generalized Khovanov bracket to smooth link cobordisms in R^3×I and proving functoriality up to global invertible scalars.
result Generalized Khovanov bracket is functorial up to global invertible scalars.
Simplifies fixing Khovanov homology functoriality.
problem Fixing the sign indeterminacy in Khovanov homology.
method Adjusting signs of Reidemeister and Morse moves.
result Functoriality of Khovanov homology is fixed.
Functorial properties of knot invariants under specific concordances.
problem Computing obstructions to regular Lagrangian concordances.
method Functoriality of EH class and LOSS invariant under Lagrangian concordances in Weinstein cobordisms.
result Computable obstructions to regular Lagrangian concordances.
Proves Khovanov homology functoriality and positivity for gl2 webs.
problem Proving functoriality and positivity of Khovanov homology.
method Categorical proof using linear complexes stability.
result Strong positivity result for surface skein algebras.
Clarifies properties of Ozsvath-Szabo contact invariant.
problem Properties and functoriality of Ozsvath-Szabo contact invariant.
method Explains and proves functoriality properties of the invariant.
result Strong functoriality under Stein cobordisms.
New method for integrating singular foliations via paths.
problem Characterize holonomy and fundamental groupoids of singular foliations.
method Quotient of an infinite dimensional space of paths, extending classical construction for regular foliations.
result Characterization of holonomy and fundamental groupoids of singular foliations.
Unified framework for proving functoriality of various link homology theories.
problem Functoriality and Reidemeister invariance of link homology theories.
method Unified framework leveraging Khovanov homology and various link homology theories.
result Stronger functoriality results by avoiding spectral sequences.
There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fit…
Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion…
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
Anomaly in free fermion theory revealed in functorial field theory.
problem Chiral anomaly in the free fermion theory.
method Detailed construction of anomaly theory as a functor.
result The anomaly theory assigns elements of complex line to manifolds.
We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of…
Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
Extends odd Khovanov bracket to link cobordisms and proves functoriality up to sign.
problem Proving functoriality of odd Khovanov homology up to sign.
method Extending odd Khovanov bracket to link cobordisms and proving functoriality up to sign.
result Functoriality of odd Khovanov homology up to sign.
Functorial compactification of vector spaces defined as manifolds with corners.
problem Compactification of vector spaces functorial under linear maps.
method Definition of manifolds with corners and b-maps, application of iterated blow-up theory.
result Criterion for lifted maps to be b-fibrations, identification of restrictions to boundary hypersurfaces.
A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. …
Quantum field theory uses Lorentzian bordisms to describe time evolution.
problem Describing the time evolution of quantum field theories.
method Defines a functorial field theory on Lorentzian bordism pseudo-category.
result Lorentzian bordisms naturally arise in algebraic quantum field theory.
Alternative proof of Khovanov's up-to-sign functoriality for odd Khovanov homology.
problem Functoriality of Khovanov's odd Khovanov homology.
method Extending Hochschild (co)homology to quasi-associative algebras and applying it to Khovanov's homology.
result First proof of functoriality of Naisse and Putyra's tangle theory up to unit.
Study normal bundle and deformation to get new pushforward maps.
problem Construct pushforward maps in various homology theories.
method Use deformation Lie groupoids to construct pushforward maps.
result Functoriality of pushforward maps recovers and generalizes previous cases.
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
New trisection concept for non-closed 4-manifolds connects to group theory.
problem Defining and analyzing trisections of non-closed 4-manifolds.
method Generalized group trisections to non-closed cases and established a one-to-one correspondence.
result Established a functorial relationship between relative trisections and groups.
We verify that the formula of X. Ma for the analytic torsion form of an iterated fibration implies that Lott's secondary analytic index is functorial.
This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to…
The paper describes new types of picture-valued invariants and their applications.
problem Understanding and categorizing picture-valued invariants on diagrams.
method General description and classification of derivations and functorial maps.
result Two new examples of functorial maps are introduced, including the order and lifting maps.
Establishes functoriality of Baum-Bott residues under specific conditions.
problem Functoriality of Baum-Bott residues under certain conditions.
method Establishes functoriality of Baum-Bott residues under specific conditions.
result The singular set of a foliation has dimension at least \( k-1 \).
This work draws inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carlsson and Mémoli to par…
The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.
problem Understanding composition of Lagrangian correspondences in Hilbert spaces.
method Study of Lagrangian correspondences, construction of a category, and functorial field theory.
result Well-defined composition law in a category of Lagrangian correspondences.
In this paper, we define the equivariant eta form of Bismut-Cheeger for a compact Lie group and establish a formula about the functoriality of equivariant eta forms with respect to the composition of two submersions.
Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.
This paper introduces complex Chern-Simons bundles in families setting and proves their crystalline nature.
problem Characterizing projective structures of Riemann surfaces and establishing holomorphic torsion formulas.
method Develops a formalism for direct images of characteristic classes, uses deformation theory of harmonic maps, and relies on non-abelian Hodge theory.
result Establishes the crystalline nature of the relative complex Chern-Simons bundle and its holomorphic extension.
Functoriality proved for higher rho invariants of elliptic operators.
problem Computing higher rho invariants of elliptic operators.
method Functoriality proved through finite-propagation argument.
result Maximal higher rho invariants behave functorially under quotient maps.
We disprove the conjecture of M. Khovanov (math.QA/9908171) on the functoriality of his link homology with polynomial coefficients. This is in contrast to the case of integer coefficients, where functoriality was proved in math.GT/0206303 .
Functorial correspondence found between logarithmic connections and abelian connections.
problem Establishing a relationship between logarithmic sl2-connections and abelian connections. method Constructing inverse functors and a canonical cocycle.
result Functorial correspondence proved between logarithmic sl2-connections and abelian connections. Paper establishes equivalence between algebraic and functorial QFTs.
problem Challenges in globally hyperbolic Lorentzian bordisms and their impact on field theories.
method Introduces a new concept of bordisms and pseudo-operads to address challenges, defining FQFTs as pseudo-multifunctors.
result Equivalence theorem between globally hyperbolic Lorentzian FQFTs and AQFTs.
The Rips complex at scale r is homotopy equivalent to the nerve of a cover of diameter r.
problem Reconstructing spaces using Rips complexes and covers.
method Functorial Dowker-Nerve Diagram, homotopy equivalence, cover of diameter r.
result General framework for reconstructing spaces by Rips complexes.
To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first …
Functorial maps and weak parities are equivalent descriptions of rules of substitution virtual crossings for classical in diagrams of a knot in a way compatible with Reidemeister moves. We introduce the notion of maximal weak parity and describe it for knots in a given closed oriented surface. This weak parity defines …
Link Floer homology maps are constructed for link cobordisms.
problem Link Floer homology invariance and functoriality.
method Defined cobordism maps on a curved chain homotopy type invariant.
result Invariance of constructed cobordism maps proved.
Refines a tangle invariant using XC-algebras.
problem Building a refined tangle invariant using XC-algebras.
method Constructing a canonical strict monoidal functor that refines the Kerler-Kauffman-Radford invariant.
result Preserves the braiding, twist, and open trace.
We discuss two kinds of functorial prolongations of the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. We study the prolongation of vector fields in both cases and we prove that the bracket is preserved. Our proof is based on several new res…
Develops a new framework for anomaly description in quantum field theories.
problem Anomalies in quantum field theories.
method Extended functorial field theories and symmetric monoidal bicategories.
result Explicit construction of anomaly 2-cocycles and 't Hooft anomalies.
Let F be a field and let G⊂F∖{0} be a multiplicative subgroup. We consider the category CobG of 3-dimensional cobordisms equipped with a representation of their fundamental group in G, and the category VectF,±G of F-linear maps defin…