Cluster structures for 2-Calabi-Yau categories and unipotent groups

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised, especially Chapter III replaces the old Chapter III and I

Universal homogeneous constraint structures and the hom-equivalence classes of weakly oligomorphic structures

We derive a new sufficient condition for the existence of {\omega}-categorical universal structures in classes of relational structures with constraints, augmenting results by Cherlin, Shelah, Chi, and Hubi\v{c}ka and Ne\v{s}et\v{r}il. Using this result we show that the hom-equivalence class of any countable weakly oligomorphic structure has up to isomorphism a unique model-complete smallest and greatest element, both of which are {\omega}-categorical. As the main tool we introduce the category of constraint structures, show the existence of universal homogeneous objects, and study their automorphism groups. All constructions rest on a category-theoretic version of Fra\"iss\'e's Theorem due to Droste and G\"obel. We derive sufficient conditions for a comma category to contain a universal homogeneous object. This research is motivated by the observation that all countable models of the theory of a weakly oligomorphic structure are hom-equivalent---a result akin to (part of) the Ryll-Nardzewski Theorem.Comment: 25 page

Restricted Priestley dualities and discriminator varieties

Anyone who has ever worked with a variety~$\boldsymbol{\mathscr{A}}$ of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one---but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category $\boldsymbol{\mathscr{X}}$ of such a variety, we give a characterisation, in terms of $\boldsymbol{\mathscr{X}}$, of finitely generated discriminator subvarieties of~$\boldsymbol{\mathscr{A}}$. As a first application of our characterisation, we give a new proof of Sankappanavar's characterisation of finitely generated discriminator varieties of distributive double p-algebras. A substantial portion of the paper is devoted to the application of our results to Cornish algebras. A Cornish algebra is a bounded distributive lattice equipped with a family of unary operations each of which is either an endomorphism or a dual endomorphism of the bounded lattice. They are a natural generalisation of Ockham algebras, which have been extensively studied. We give an external necessary-and-sufficient condition and an easily applied, completely internal, sufficient condition for a finite set of finite Cornish algebras to share a common ternary discriminator term and so generate a discriminator variety. Our results give a characterisation of discriminator varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen and Pitkethly's characterisation of quasi-primal Ockham algebras

Metric abstract elementary classes as accessible categories

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete $\aleph_1$-directed colimits and concrete monomorphisms. More broadly, we define a notion of $\kappa$-concrete AEC---an AEC-like category in which only the $\kappa$-directed colimits need be concrete---and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [LR] yield a proof that any categorical mAEC is $\mu$-d-stable in many cardinals below the categoricity cardinal.Comment: v2: changed terminology. v3: tightened inequalities. v4: clarifying notes added. v5: referee's comments incorporated, with substantial improvement

Multidimensional exact classes, smooth approximation and bounded 4-types

In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class ($R$-mec), a special kind of multidimensional asymptotic class ($R$-mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatisation [14] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language $\mathcal{L}$ and any positive integer $d$ the class $\mathcal{C}(\mathcal{L},d)$ of all finite $\mathcal{L}$-structures with at most $d$ 4-types is a polynomial exact class in $\mathcal{L}$, where a polynomial exact class is a multidimensional exact class with polynomial measuring functions

Optical Substructures in 48 Galaxy Clusters

We analyze the presence of substructures in a set of 48 galaxy clusters, by using galaxy positions and redshifts. We use a multi-scale analysis which couples kinematical estimators with the wavelet transform. 14% of our clusters are strongly substructured (i.e. they are bimodal or complex) and 24% of the remaining unimodal clusters contain substructures at small scales. Thus, in substantial agreement with previous studies, about one third of clusters show substructures. In unimodal clusters the presence of substructures does not affect the estimates of both virial masses and velocity dispersions, which are generally in good agreement with the X-ray temperatures. Thus, unimodal clusters are not too far from a status of dynamical equilibrium. On the contrary, velocity dispersions and masses for some bimodal or complex clusters strongly depend on whether they are treated as single systems or as sums of different clumps and X-ray temperatures and velocity dispersions may be very different.Comment: 24 pages, 5 bitmapped eps figures, 7 tables, USE LaTeX2e !! Astrophysical Journal, in pres

Universal abstract elementary classes and locally multipresentable categories

We exhibit an equivalence between the model-theoretic framework of universal classes and the category-theoretic framework of locally multipresentable categories. We similarly give an equivalence between abstract elementary classes (AECs) admitting intersections and locally polypresentable categories. We use these results to shed light on Shelah's presentation theorem for AECs.Comment: 14 pages. Some typos remove

Higher-dimensional analogues of K3 surfaces

In the first part we survey some of the known results and conjectures on compact Hyperkaehler (HK) manifolds. In the second part we presents a program which aims to show that HK four-folds whose second cohomology (with 4-tuple cup-product) is isomorphic to that of the Hilbert square of a K3 enjoy many of the beautiful properties of K3 surfaces, in particular they should form a single deformations class.Comment: The paper originated from a talk given at MSRI during the special program in Algebraic Geometry (2009)

SUPPLEMENTED MORPHISMS

In the present paper, left R-modules M and N are studied under the assumptions that HomR(M,N) is supplemented. It is shown that Hom(M,N) is (⊕, G*, amply)-supplemented if and only if N is (⊕, G*, amply)-supplemented. Some applications to cosemisimple modules, refinable modules and UCC-modules are presented. Finally, the relationship between the Jacobson radical J[M,N] of HomR(M,N) and HomR(M,N) is supplemented are investigated. Let M be a finitely generated, self-projective left R-module and N ∈ Gen(M). We show that if Hom(M,N) is supplemented and N has GD2 then Hom(M,N)/J(M,N) is semisimple as a left EM-module

Motivic toposes

We present a research programme aimed at constructing classifying toposes of Weil-type cohomology theories and associated categories of motives, and introduce a number of notions and preliminary results already obtained in this direction. In order to analyze the properties of Weil-type cohomology theories and their relations, we propose a framework based on atomic two-valued toposes and homogeneous models. Lastly, we construct a syntactic triangulated category whose dual maps to the derived categories of all the usual cohomology theories.Comment: 41 page