The Pure Virtual Braid Group Is Quadratic

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of such a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a (not necessarily homomorphic) universal finite type invariant.Comment: 53 pages, 15 figures. Some clarifications added and inaccuracies corrected, reflecting suggestions made by the referee of the published version of the pape

The fibre of a pinch map in a model category

In the category of pointed topological spaces, let F be the homotopy fibre of the pinching map X ∪ CA → X ∪ CA/ X from the mapping cone on a cofibration A → X onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497–520, 1973) proved that F is weakly homotopy equivalent to the reduced product (X, A)∞. In this paper we prove an analogue of this phenomenon in a model category, under suitable conditions including a cube axiom.Web of Scienc

The Dold-Kan Correspondence and Coalgebra Structures

By using the Dold-Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected coalgebras and prove a Quillen equivalence between them.Comment: 24 pages. Accepted by the Journal of Homotopy and Related Structures. Online 28 November 201

Understanding the small object argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.Comment: 42 pages; supersedes the earlier arXiv preprint math/0702290; v2: final journal version, minor corrections onl

Rediscovering the value of families for psychiatric genetics research

As it is likely that both common and rare genetic variation are important for complex disease risk, studies that examine the full range of the allelic frequency distribution should be utilized to dissect the genetic influences on mental illness. The rate limiting factor for inferring an association between a variant and a phenotype is inevitably the total number of copies of the minor allele captured in the studied sample. For rare variation, with minor allele frequencies of 0.5% or less, very large samples of unrelated individuals are necessary to unambiguously associate a locus with an illness. Unfortunately, such large samples are often cost prohibitive. However, by using alternative analytic strategies and studying related individuals, particularly those from large multiplex families, it is possible to reduce the required sample size while maintaining statistical power. We contend that using whole genome sequence (WGS) in extended pedigrees provides a cost-effective strategy for psychiatric gene mapping that complements common variant approaches and WGS in unrelated individuals. This was our impetus for forming the “Pedigree-Based Whole Genome Sequencing of Affective and Psychotic Disorders” consortium. In this review, we provide a rationale for the use of WGS with pedigrees in modern psychiatric genetics research. We begin with a focused review of the current literature, followed by a short history of family-based research in psychiatry. Next, we describe several advantages of pedigrees for WGS research, including power estimates, methods for studying the environment, and endophenotypes. We conclude with a brief description of our consortium and its goals.This research was supported by National Institute of Mental Health grants U01 MH105630 (DCG), U01 MH105634 (REG), U01 MH105632 (JB), R01 MH078143 (DCG), R01 MH083824 (DCG & JB), R01 MH078111 (JB), R01 MH061622 (LA), R01 MH042191 (REG), and R01 MH063480 (VLN).UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigación en Biología Celular y Molecular (CIBCM)UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Biologí