Four problems regarding representable functors

Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\mathcal M}$ the category of left $C$-comodules. The category ${\bf Rep}\, ( {}_{R}^C{\mathcal M}, {}_{S}{\mathcal M} )$ of all representable functors ${}_{R}^C{\mathcal M} \to {}_{S}{\mathcal M}$ is shown to be equivalent to the opposite of the category ${}_{R}^C{\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give necessary and sufficient conditions for the induction functor $U\otimes_R - : {}_{R}^C\mathcal{M} \to {}_{S}\mathcal{M}$ to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings.Comment: 16 pages, the second versio

Self-Reported Memory for Abuse Depends Upon Victim-Perpetrator Relationship

Article copies available for a fee from The Haworth Document Delivery Service: 1-800-342-9678. E-mail address: Website: (c) 2001 by The Haworth Press, Inc. All rights reserved.We present preliminary results from the Betrayal Trauma Inventory (BTI) testing predictions from betrayal trauma theory (Freyd, 1994, 1996, in press) about the relationship between amnesia and betrayal by a caregiver. The BTI assesses trauma history using behaviorally defined events in the domains of sexual, physical, and emotional childhood abuse, as well as other lifetime traumatic events. When participants endorse an abuse experience, follow-up questions assess a variety of factors including memory impairment and perpetrator relationship. Preliminary results support our prediction that abuse perpetrated by a caregiver is related to less persistent memories of abuse. This relationship is significant for sexual and physical abuse. Regression analyses revealed that age was not a significant predictor of memory impairment and that duration of abuse could not account for the findings

Virtual Khovanov homology using cobordisms

We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathematica based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology.Comment: 78 pages, lots of figures, lots of typos, rewritten version, merged with arXiv:1212.0185, added referee's suggestions, to appear in J. Knot Theor. Rami

A Coboundary Morphism For The Grothendieck Spectral Sequence

Given an abelian category $\mathcal{A}$ with enough injectives we show that a short exact sequence of chain complexes of objects in $\mathcal{A}$ gives rise to a short exact sequence of Cartan-Eilenberg resolutions. Using this we construct coboundary morphisms between Grothendieck spectral sequences associated to objects in a short exact sequence. We show that the coboundary preserves the filtrations associated with the spectral sequences and give an application of these result to filtrations in sheaf cohomology.Comment: 18 page

Sound and complete axiomatizations of coalgebraic language equivalence

Coalgebras provide a uniform framework to study dynamical systems, including several types of automata. In this paper, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalised powerset construction that determinises coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor $FT$, where $T$ is a monad describing the branching of the systems (e.g. non-determinism, weights, probability etc.), has as a quotient the rational fixpoint of the "determinised" type functor $\bar F$, a lifting of $F$ to the category of $T$-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain non-deterministic automata, where we recover Rabinovich's sound and complete calculus for language equivalence.Comment: Corrected version of published journal articl

Division, adjoints, and dualities of bilinear maps

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The bilinear division maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, nonassociative division rings can be studied within this framework. This also corrects an error in an earlier pre-print; see Remark 2.11

2-Vector Spaces and Groupoids

This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation--a weak functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.Comment: 44 pages, 5 figures - v2 adds new theorem, significant changes to proofs, new sectio

University Crime Alerts: Do They Contribute to Institutional Betrayal and Rape Myths?

Universities are mandated by the Clery Act (20 USC § 1092(f)) to publicize the occurrence of certain campus crimes. Many universities rely on “Crime Alert” emails to quickly and effectively communicate when a crime has occurred. However, communications of sexual crimes are often narrow (e.g., limited to stranger-perpetrated crimes) and misleading (e.g., containing safety tips that are not applicable to most types of sexual violence). The current paper presents the results of two studies that test the effects of reading crime alert emails on subsequent endorsement of rape myths and institutional betrayal. In Study 1, participants read a typical crime alert email describing a stranger-perpetrated crime, an alternative email describing an acquaintance-perpetrated crime, or a control email describing an event unrelated to interpersonal violence. Men were significantly more likely to endorse rape myths than were women in the control condition, but not in the typical or alternative email condition. In addition, results from Study 1 indicate that issuing crime alert emails following stranger-perpetrated sexual violence leads to a sense of institutional betrayal among students who have experienced acquaintance-perpetrated violence. In Study 2, participants read a typical crime alert email or an alternative digest email. Participants who read the typical email reported higher rape myth acceptance, but not institutional betrayal, than those who read the digest email. There were also significant gender differences in student opinions of each email that suggest the digest email format may serve as a useful tool for engaging male students in the issue of campus sexual violence. Taken together, these studies provide converging evidence that university communication regarding sexual violence can either perpetuate or positively influence attitudes towards sexual violence