Functorial Approach Research Articles

A functorial approach to the stability of vector bundles

On a normal projective variety the locus of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable bundles that remain μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles. As an application we obtain a very different behaviour of the étale fundamental group in positive vs. characteristic 0.

A functorial approach to rank functions on triangulated categories

Abstract We study rank functions on a triangulated category 𝒞 via its abelianisation mod ⁡ C \operatorname{mod}\mathcal{C} . We prove that every rank function on 𝒞 can be interpreted as an additive function on mod ⁡ C \operatorname{mod}\mathcal{C} . As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category Mod ⁡ C \operatorname{Mod}\mathcal{C} . We study the connection between rank functions and functors from 𝒞 to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case C = T c \mathcal{C}=\mathcal{T}^{c} for a compactly generated triangulated category 𝒯, this connection becomes particularly nice, providing a link between rank functions on 𝒞 and smashing localisations of 𝒯. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in 𝒯. Finally, if C = per ⁡ ( A ) \mathcal{C}=\operatorname{per}(A) for a differential graded algebra 𝐴, we classify homological epimorphisms A → B A\to B with per ⁡ ( B ) \operatorname{per}(B) locally finite via special rank functions which we call idempotent.

Homotopy relative Rota-Baxter lie algebras, triangular 𝐿_{∞}-bialgebras and higher derived brackets

We describe L ∞ L_\infty -algebras governing triangular L ∞ L_\infty -bialgebras and homotopy relative Rota-Baxter Lie algebras and establish a map between them. Our formulas are based on a functorial approach to Voronov’s higher derived brackets construction which is of independent interest.

A functorial approach to regular homomorphisms

Classically, regular homomorphisms have been defined as a replacement for Abel--Jacobi maps for smooth varieties over an algebraically closed field. In this work, we interpret regular homomorphisms as morphisms from the functor of families of algebraically trivial cycles to abelian varieties and thereby define regular homomorphisms in the relative setting, e.g., families of schemes parameterized by a smooth variety over a given field. In that general setting, we establish the existence of an initial regular homomorphism, going by the name of algebraic representative, for codimension-2 cycles on a smooth proper scheme over the base. This extends a result of Murre for codimension-2 cycles on a smooth projective scheme over an algebraically closed field. In addition, we prove base change results for algebraic representatives as well as descent properties for algebraic representatives along separable field extensions. In the case where the base is a smooth variety over a subfield of the complex numbers we identify the algebraic representative for relative codimension-2 cycles with a subtorus of the intermediate Jacobian fibration which was constructed in previous work. At the heart of our descent arguments is a base change result along separable field extensions for Albanese torsors of separated, geometrically integral schemes of finite type over a field.

Real Forms of Complex Lie Superalgebras and Supergroups

We investigate the notion of real form of complex Lie superalgebras and supergroups, both in the standard and graded version. Our functorial approach allows most naturally to go from the superalgebra to the supergroup and retrieve the real forms as fixed points, as in the ordinary setting. We also introduce a more general notion of compact real form for Lie superalgebras and supergroups, and we prove some existence results for Lie superalgebras that are simple contragredient and their associated connected simply connected supergroups.

Symmetry defects and orbifolds of two-dimensional Yang–Mills theory

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

A functorial approach to monomorphism categories for species I

We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalized species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit computations as in the case of Ringel and Schmidmeier.

Torsion Pairs and Ringel Duality for Schur Algebras

Let A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

Fusion of interfaces in Landau-Ginzburg models: a functorial approach

We investigate the fusion of B-type interfaces in two-dimensional supersymmetric Landau-Ginzburg models. In particular, we propose to describe the fusion of an interface in terms of a fusion functor that acts on the category of modules of the underlying polynomial rings of chiral superfields. This uplift of a functor on the category of matrix factorisations simplifies the actual computation of interface fusion. Besides a brief discussion of minimal models, we illustrate the power of this approach in the SU(3)/U(2) Kazama-Suzuki model where we find fusion functors for a set of elementary topological defects from which all rational B-type topological defects can be generated.

A Functorial Approach to Gabriel k-quiver Constructions for Coalgebras and Pseudocompact Algebras

We define the path coalgebra and Gabriel quiver constructions as functors between the category of $k$-quivers and the category of pointed $k$-coalgebras, for $k$ a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel $k$-quiver functor is left adjoint to the corresponding path coalgebra functor. We dualize, obtaining adjoint pairs of functors (contravariant and covariant) for pseudocompact algebras. Using these tools we describe precisely to what extent presentations of coalgebras and algebras in terms of path objects are unique, giving an application to homogeneous algebras.

A functorial approach to categorical resolutions

Using a relative version of Auslander's formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories, introduced in representation theory by Auslander, to categorical resolutions.

Functorial hierarchical clustering with overlaps

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 partition-based clustering, and the Isbell/Dress school’s study of injective envelopes. Carlsson and Mémoli introduce the idea of viewing clustering methods as functors from a category of metric spaces to a category of clusters, with functoriality subsuming many desirable properties. Our first series of results extends their theory of functorial clustering schemes to methods that allow overlapping clusters in the spirit of Jardine and Sibson. This obviates some of the unpleasant effects of chaining that occur, for example with single-linkage clustering. We prove an equivalence between these general overlapping clustering functors and projections of weight spaces to what we term clustering domains, by focusing on the order structure determined by the morphisms. As a specific application of this machinery, we are able to prove that there are no functorial projections to cut metrics, or even to tree metrics. Finally, although we focus less on the construction of clustering methods (clustering domains) derived from injective envelopes, we lay out some preliminary results, that hopefully will give a feel for how the third leg of the stool comes into play.

Poisson–Riemannian geometry

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter λ , using a functorial approach. This leads us to field equations of ‘Poisson–Riemannian geometry’ between the classical metric, the Poisson bracket and a certain Poisson-compatible connection needed as initial data for the quantisation of the differential structure. We use such data to define a functor Q to O ( λ 2 ) from the monoidal category of all classical vector bundles equipped with connections to the monoidal category of bimodules equipped with bimodule connections over the quantised algebra. This is used to ‘semiquantise’ the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection in the sense of constructing a noncommutative geometry to O ( λ 2 ) . We solve our field equations for the Schwarzschild black-hole metric under the assumption of spherical symmetry and classical dimension, finding a unique solution and the necessity of nonassociativity at order λ 2 , which is similar to previous results for quantum groups. The paper also includes a nonassociative hyperboloid, nonassociative fuzzy sphere and our previously algebraic bicrossproduct model.

The twofold way of super holonomy

Abstract There are two different notions of holonomy in supergeometry, the supergroup introduced by Galaev and our functorial approach motivated by super Wilson loops. Either theory comes with its own version of invariance of vectors and subspaces under holonomy. By our first main result, the Twofold Theorem, these definitions are equivalent. Our proof is based on the Comparison Theorem, our second main result, which characterises Galaev’s holonomy algebra as an algebra of coefficients, building on previous results. As an application, we generalise some of Galaev’s results to S-points, utilising the holonomy functor. We obtain, in particular, a de Rham–Wu decomposition theorem for semi-Riemannian S-supermanifolds.

From K-Nets to PK-Nets: A Categorical Approach

FROM K-NETS TO PK-NETS: A CATEGORICAL APPROACH ALEXANDRE POPOFF CARLOS AGON MORENO ANDREATTA ANDRÉE EHRESMANN 1. INTRODUCTION RANSFORMATIONAL APPROACHES have a long tradition in formalized music analysis in the American as well as in the European tradition. Since the publication of pioneering work by David Lewin (1987) and Guerino Mazzola (1990), this paradigm has become an autonomous field of study making use of more and more sophisticated mathematical tools, ranging from group theory to categorical methods via graphtheoretical constructions (Nolan 2007). Within the transformational approach, Klumpenhouwer networks (henceforth K-nets) are prototypical examples of music-theoretical constructions unifying the three domains we just mentioned, since they provide a description of the inner structure of chords by focusing on the transformations between their elements rather than on the elements themselves. For this reason, K-nets T 6 Perspectives of New Music represent a complementary approach to the traditional set-theoretical one with respect to the problem of chord enumeration and classification. One of the main interests for the “working mathemusician” lies in their deep connections with some common constructions used in category theory. In fact, following Mazzola’s original intuition on the relevance of the categorical approach to the formalization of musical structures and process, Klumpenhouwer networks seem suitable for music-theoretical investigations making use of category theory, since they are based on concepts (such as isographies) and principles (such as the recursive network construction) which are naturally grasped by the functorial approach. However, as we have suggested elsewhere (Popoff et al. 2015), although K-nets and, more generally, group action-based theoretical constructions, such as Lewin’s “Generalized Interval Systems ” (GIS), are naturally described in terms of categories and functors, the categorical approach to transformational theory remains relatively marginal with respect to the major trend in the math-music community (Mazzola 2002; Lavelle, unpublished paper; Fiore 2011; Popoff, submitted paper). Following Lewin’s (1990) and Klumpenhouwer’s (1991) original group-theoretical descriptions, theoretical studies have mostly focused until now on the automorphisms of the T/I group or of the more general T/M affine group (Lewin 1990; Klumpenhouwer 1998). This enables one to define the main notions of positive and negative isographies, notions which can easily be extended by taking into account the affine group on ℤ12, together with high-order isographies . Since a prominent feature of K-nets is their ability to instantiate an in-depth multi-level model of musical structure, category theory seems nowadays the most suitable mathematical framework to capture this recursive potentiality of the graph-theoretical construction (Mazzola et al. 2006) or to study creativity; e.g., in music (Andreatta et al. 2013). From a graph-theoretical perspective, a K-net is a directed graph (also called digraph) where the vertices (or nodes) consist of pitchclasses (or pitch-class sets) and the edges (or arrows) are elements of the T/I group (or, in a more general setting, of the T/M affine group). As an example, we represent two K-nets in Example 1. In addition, these K-nets are 〈T2〉-isographic, in the sense that every edge of the form Tp is sent to Tp and every edge of the form Ip is sent to Ip+2. In the general case, what music theorists call the “path consistency condition” (Hook 2007) is nothing else than the composition and associativity law of morphisms, together with the definition of functors in the categorical framework. Moreover, all K-nets correspond to commutative diagrams in category theory, where the term “diagram” is taken here in a naive sense (the technical concept of a diagram will be introduced in Section 5.1.) From K-Nets to PK-Nets: A Categorical Approach 7 Commutative diagrams, together with the notion of isography as an algebraic relation between K-nets which is independent of the content of the nodes, clearly suggest that the categorical approach is the natural one for the study of any kind of networks. Moreover, category theory shows how to go beyond the K-nets commonly used in transformational analysis by defining diagrams which do not necessarily have this property and by extending the isographic relation to different levels, by considering transformations between networks...

On non-realization results and conjectures of N. Kuhn

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod-p singular cohomology of a space, for p a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes’ mapping space theorem. The second (for the prime 2) is based on an analysis of the -1 and -2 columns of the Eilenberg–Moore spectral sequence, and of the associated extension. In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between Fp-vector spaces. The second result in particular exhibits the power of the functorial approach.

A construction of [formula omitted]-adic modular forms

TextThe classical theory of p-adic (elliptic) modular forms arose in the 1970ʼs in the work of J.-P. Serre [Se1] who took p-adic limits of the q-expansions of these forms. It was soon expanded by N. Katz [Ka1] with a more functorial approach. In the late 1970ʼs, the theory of modular forms associated to Drinfeld modules was born in analogy with elliptic modular forms [Go1,Go2]. The associated expansions at ∞ are quite complicated with no obvious limits at finite primes v. Recently, A. Petrov [Pe1] showed that there is an intermediate expansion at ∞ called the “A-expansion,” and constructed families of cusp forms with such expansions. It is our purpose in this note to show that Petrovʼs results also lead to interesting v-adic cusp forms à la Serre. Moreover the existence of these forms allows us to readily conclude a mysterious decomposition of the associated Hecke action. VideoFor a video summary of this paper, please click here or visit http://youtu.be/xzezUI7-3yc.

2-Dimensional Directed Type Theory

Recent work on higher-dimensional type theory has explored connections between Martin-Löf type theory, higher-dimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality—for example, taking IdSetA B to be the isomorphisms between A and B. The crucial observation is that all of the familiar type and term constructors can be equipped with a functorial action that describes how they preserve such proofs. The key benefit of higher-dimensional type theory is that programmers and mathematicians may work up to isomorphism and higher equivalence, such as equivalence of categories.In this paper, we consider a further generalization of higher-dimensional type theory, which associates each type with a directed notion of transformation between its elements. Directed type theory accounts for phenomena not expressible in symmetric higher-dimensional type theory, such as a universe set of sets and functions, and a type Ctx used in functorial abstract syntax. Our formulation requires two main ingredients: First, the types themselves must be reinterpreted to take account of variance; for example, a Π type is contravariant in its domain, but covariant in its range. Second, whereas in symmetric type theory proofs of equivalence can be internalized using the Martin-Löf identity type, in directed type theory the two-dimensional structure must be made explicit at the judgemental level. We describe a 2-dimensional directed type theory, or 2DTT, which is validated by an interpretation into the strict 2-category Cat of categories, functors, and natural transformations. We also discuss applications of 2DTT for programming with abstract syntax, generalizing the functorial approach to syntax to the dependently typed and mixed-variance case.

On a new compactification of moduli of vector bundles on a surface. III: Functorial approach

A new compactification for the scheme of moduli for Gieseker-stable vector bundles with prescribed Hilbert polynomial on the smooth projective polarized surface is constructed. We work over the field of characteristic zero. Families of locally free sheaves on the surface are completed with locally free sheaves on schemes which are modifications of . The Gieseker-Maruyama moduli space has a birational morphism onto the new moduli space. We propose the functor for families of pairs 'polarized scheme-vector bundle' with moduli space of such type.Bibliography: 16 titles.

Derived functors of nonadditive functors and homotopy theory

We develop a functorial approach to the study of the homotopy groups of spheres and Moore spaces $M(A,n)$, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie algebra functors, as well as of all the main cubical functors (such as the degree 3 component $SP^3$ of the symmetric algebra functor). As an illustration of this method, we retrieve in a purely algebraic manner the 3-torsion component of the homotopy groups of the 2-sphere up to degree 14, and give a unified presentation of homotopy groups $\pi_i(M(A,n))$ for small values of both $n$ and $i$.