arXiv subjects
Publications and source records attributed to Ettore Aldrovandi.
We extend McCarthy's stabilization construction to exact $\infty$-categories. This is achieved by constructing, for any functor from exact $\infty$-categories to a fixed stable $\infty$-category $\mathcal{A}$, a coherent chain complex in $\mathcal{A}$ that is an immediate generalization of Mac Lane's cubical $Q$-complex computing the stable homology of abelian groups.
We study monoidal 2-categories and bicategories in terms of categorical extensions and the cohomological data they determine in appropriate cohomology theories with coefficients in Picard groupoids. In particular, we analyze the hierarchy of possible commutativity conditions in terms of progressive stabilization of these data. We also show that monoidal structures on bicategories give rise to biextensions of a pair of (abelian) groups by a Picard groupoid, and that the progressive vanishing of obstructions determined by the tower of commutative structures corresponds to appropriate symmetry conditions on these biextensions. In the fully symmetric case, which leads us fully into the stable range, we show how our computations can be expressed in terms of the cubical Q-construction underlying MacLane (co)homology.
We extend Deligne's notion of determinant functor to tensor triangulated categories. Specifically, to account for the multiexact structure of the tensor, we define a determinant functor on the 2-multicategory of triangulated categories and we provide a multicategorical version of the universal determinant functor for triangulated categories, whose multiexactness properties are conveniently captured by a certain complex modeled by cubical shapes, which we introduce along the way. We then show that for a tensor triangulated category whose tensor admits a Verdier structure the resulting determinant functor takes values in a categorical ring.
Let $π: X \to S$ be a family of smooth projective curves, and let $L$ and $M$ be a pair of line bundles on $X$. We show that Deligne's line bundle $\langle{L,M}\rangle$ can be obtained from the $\mathcal{K}_2$-gerbe $G_{L,M}$ constructed in a previous work by the authors via an integration along the fiber map for gerbes that categorifies the well known one arising from the Leray spectral sequence of $π$. Our construction provides a full account of the biadditivity properties of $\langle {L,M}\rangle$. The functorial description of the low degree maps in the Leray spectral sequence for $π$ we develop are of independent interest, and along the course we provide an example of their application to the Brauer group.
We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.
We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, these data determine a bimonoidal functor. We extend this result to $n$-variables, and prove that, in a manner analogous to that of butterflies, these multi-extensions can be composed. This is phrased in terms of a multilinear functor calculus in a bicategory. As an application, we study a bimonoidal category or stack, treating the multiplicative structure as a bimonoidal functor with respect to the additive one. In the context of the multilinear functor calculus, we view the bimonoidal structure as an instance of the general notion of pseudo-monoid. We show that when the structure is ring-like, i.e. the pseudo-monoid is a stack whose fibers are categorical rings, we can recover the classification by the third Mac Lane cohomology of a ring with values in a bimodule.
We define higher categorical invariants (gerbes) of codimension two algebraic cycles and provide a categorical interpretation of the intersection of divisors on a smooth proper algebraic variety. This generalization of the classical relation between divisors and line bundles furnishes a new perspective on the Bloch-Quillen formula.
The weak units of strict monoidal 1- and 2-categories are already defined. In this paper, we define them for group-like 1- and 2-stacks. We show that they form a contractible Picard 1- and 2-stack, respectively. We give their cohomological description which provides for these stacks a representation by complexes of sheaves of groups. Later, we extend the discussion to the monoidal case.
We show that $\mathit{ann}$-categories admit a presentation by crossed bimodules, and prove that morphisms between them can be expressed by special kinds spans between the presentations. More precisely, we prove the groupoid of morphisms between two $\mathit{ann}$-categories is equivalent to that of bimodule butterflies between the presentations. A bimodule butterfly is a specialization of a butterfly, i.e. a special kind of span or fraction, between the underlying complexes
We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We re-interpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a crossed module. Our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the "fraction" (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at the cocycle level. In addition, we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions.
Weak morphisms of non-abelian complexes of length 2, or crossed modules, are morphisms of the associated 2-group stacks, or gr-stacks. We present a full description of the weak morphisms in terms of diagrams we call butterflies. We give a complete description of the resulting bicategory of crossed modules, which we show is fibered and biequivalent to the 2-stack of 2-group stacks. As a consequence we obtain a complete characterization of the non-abelian derived category of complexes of length 2. Deligne's analogous theorem in the case of Picard stacks and abelian sheaves becomes an immediate corollary. Commutativity laws on 2-group stacks are also analyzed in terms of butterflies, yielding new characterizations of braided, symmetric, and Picard 2-group stacks. Furthermore, the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non-abelian cohomology, removing a preexisting fibration condition on the coefficients short exact sequence.
We define 2-gerbes bound by complexes of braided group-like stacks. We prove a classification result in terms of hypercohomology groups with values in abelian crossed squares and cones of morphisms of complexes of length 3. We give an application to the geometric construction of certain elements in Hermitian Deligne cohomology groups.
We observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe which is closer to the familiar one for line bundles and does not rely on an explicit ``reduction of the structure group.'' Analogously to the case of holomorphic line bundles, a uniqueness property for the connective structure compatible with the hermitian-holomorphic structure on a gerbe is also proven. Similar results are proved for 2-gerbes as well. We then show the hermitian structures so defined propagate to a class of higher tame symbols previously considered by Brylinski and McLaughlin, which are thus found to carry corresponding hermitian-holomorphic structures. Therefore we obtain an alternative characterization for certain higher Hermitian holomorphic Deligne cohomology groups.
We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to be a functional whose extrema are the metrics of constant negative curvature.
Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Cech-de Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bundles. We reprove the relation with the determinant of cohomology construction. Furthermore, if we specialize the covering to the one provided by a Kleinian uniformization (thereby allowing possibly disconnected surfaces) the same class can be reinterpreted as the transgression of the regulator class expressed by the Bloch-Wigner dilogarithm.
We revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne cohomology class over the fundamental class of the underlying topological surface. This Deligne class is constructed by applying a descent procedure with respect to a Čech resolution of any covering map of a Riemann surface. Detailed calculations are presented in the two cases of an ordinary Čech cover, and of the universal covering map, which was used in our previous approach. We also establish a dictionary that allows to use the same formalism for different covering morphisms. The Deligne cohomology class we obtain depends on a point in the Earle-Eells fibration over the Teichmüller space, and on a smooth coboundary for the Schwarzian cocycle associated to the base-point Riemann surface. From it, we obtain a variational characterization of Hubbard's universal family of projective structures, showing that the locus of critical points for the chiral action under fiberwise variation along the Earle-Eells fibration is naturally identified with the universal projective structure.
We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation for the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus $g>1$.
We point out that the Miura transformation is related to a holomorphic foliation in a relative flag manifold over a Riemann Surface. Certain differential operators corresponding to a free field description of $W$--algebras are thus interpreted as partial connections associated to the foliation.