We generalize the notion of an anomaly for a symmetry to a noninvertible symmetry enacted by surface operators using the framework of condensation in 2-categories. Given a multifusion 2-category, ...potentially with some additional levels of monoidality, we prove theorems about the structure of the 2-category obtained by condensing a suitable algebra object. We give examples where the resulting category displays grouplike fusion rules and through a cohomology computation, and find the obstruction to condensing further to the vacuum theory. As a consequence, we show that every symmetric fusion 2-category admits a fiber 2-functor to
2
SVec
.
The higher Leray–Serre spectral sequence associated with a tower of fibrations represents a generalization of the classical Leray–Serre spectral sequence of a fibration. In this work, we present ...algorithms to compute higher Leray–Serre spectral sequences leveraging the effective homology technique, which allows to perform computations involving chain complexes of infinite type associated with interesting objects in algebraic topology. In order to develop the programs, implemented as a new module for the Computer Algebra system Kenzo, we translated the original construction of the higher Leray–Serre spectral sequence in a simplicial framework and studied some of its fundamental properties.
We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of ...the (super)symmetric power functors. We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors. By an equivalence of categories, this also provides cohomology calculations in the category of left modules over Schur superalgebras.
The aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, Déglise and Ayoub. We prove these ...t-structures possess many good properties, some analogous to those of the perverse t-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that the hearts of these t-structures provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko.
We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present ...several positive (or negative) examples associated to the vanishing theorems of Girbau, Kawamata–Viehweg, and Green–Lazarsfeld in a uniform manner and study the blow-up invariance of some classical holomorphic invariants.
Nous utiliser une approche faisceau-théorique pour obtenons une formule l'éclatement pour la cohomologie de Dolbeault groupes avec des valeurs dans le faisceau de vecteurs holomorphes sur une variété complexe compacte. Comme applications, nous présentons plusieurs exemples positifs (ou négatifs) associés á la théorèmes d'annulation de Girbau, Kawamata–Viehweg et Green–Lazarsfeld de manière uniforme et étudient l'invariance par explosion de certains invariants holomorphes.
Combining known spectral sequences with a new spectral sequence relating reduced and unreduced slN-homology yields a relationship between the Homflypt-homology of a knot and its slN-concordance ...invariants. As an application, some of the slN-concordance invariants are shown to be linearly independent.
We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first two theories were introduced respectively by Staic and Zarelua. We show in particular, ...that there is a map from exterior cohomology to symmetric cohomology which is a split monomorphism in general and an isomorphism in many cases, but not always. We introduce two spectral sequences which help to explain the relationship between these cohomology groups. As a sample application we obtain that symmetric and classical cohomologies are isomorphic for torsion free groups.
Working in a simplicial and constructive context, a new spectral system is defined that relates Serre and Eilenberg–Moore spectral sequences associated to a principal simplicial fibration. The two ...Eilenberg–Moore spectral sequences (the one where the homology of the fiber is the output, and the other where the homology of the base is computed) are used in our construction. Explicit computer programs are developed, enhancing the Kenzo computer algebra tool to implement that spectral system.
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