Obstruction Theory in Model Categories

Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.Comment: 17 pages. v3 includes improved introduction and several other minor improvement

Indicators of Woman Abuse Based on a Chart Review at a Family Practice Center

Objective: To identify demographic and health indicators of domestic violence. Design: Anonymous questionnaire survey of patients regarding violence and a chart review regarding symptoms and diagnoses. Setting: Community-based family practice residency training center in a midwestern city. Participants: Women 18 years of age or older visiting the center over a 2-month period in 1990. Of 476 eligible participants, 394 (82.7%) consented to complete the survey. Measures: A detailed, standardized measure of violence was used. Physical and psychological problems were given codes from the International Classification of Diseases, Ninth Revision (ICD-9). Results: Younger women and those separated or divorced from their partners were more likely to have been victims. Never-married women also had substantially high rates of victimization. Depression was the strongest indicator of victimization, even when controlling for demographic factors. Back pain, ulcers, headaches, and anxiety were not strong indicators of abuse. A classification analysis showed that a combination of all variables could predict lifetime injury only about half the time and violence in the past year only about 20% of the time. Conclusions: Since neither demographic nor health factors could accurately predict who had been victimized, all women need to be asked about abuse. Physicians should also keep in mind that divorced and unmarried women are often affected by abuse, either immediately or by its long term aftereffects.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/89971/1/Saunders-Hamberger-Hovey-1993-Indicators of woman abuse based on a chart review at a family practice center AFM-AMA.pd

Operations on integral lifts of K(n)

This very rough sketch is a sequel to arXiv:1808.08587; it presents evidence that operations on lifts of the functors K(n) to cohomology theories with values in modules over valuation rings of local number fields, indexed by Lubin-Tate groups of such fields, are extensions of the groups of automorphisms of the indexing group laws, by the exterior algebras on the normal bundle to the orbits of the group laws in the space of lifts.Comment: \S 2.0 hopefully less cryptic. To appear in the proceedings of the 2015 Nagoya conference honoring T Ohkawa. Comments very welcome

Morita base change in Hopf-cyclic (co)homology

In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus.Comment: Final version to appear in Lett. Math. Phy

HARP/ACSIS: A submillimetre spectral imaging system on the James Clerk Maxwell Telescope

This paper describes a new Heterodyne Array Receiver Programme (HARP) and Auto-Correlation Spectral Imaging System (ACSIS) that have recently been installed and commissioned on the James Clerk Maxwell Telescope (JCMT). The 16-element focal-plane array receiver, operating in the submillimetre from 325 to 375 GHz, offers high (three-dimensional) mapping speeds, along with significant improvements over single-detector counterparts in calibration and image quality. Receiver temperatures are $\sim$120 K across the whole band and system temperatures of $\sim$300K are reached routinely under good weather conditions. The system includes a single-sideband filter so these are SSB figures. Used in conjunction with ACSIS, the system can produce large-scale maps rapidly, in one or more frequency settings, at high spatial and spectral resolution. Fully-sampled maps of size 1 square degree can be observed in under 1 hour. The scientific need for array receivers arises from the requirement for programmes to study samples of objects of statistically significant size, in large-scale unbiased surveys of galactic and extra-galactic regions. Along with morphological information, the new spectral imaging system can be used to study the physical and chemical properties of regions of interest. Its three-dimensional imaging capabilities are critical for research into turbulence and dynamics. In addition, HARP/ACSIS will provide highly complementary science programmes to wide-field continuum studies, and produce the essential preparatory work for submillimetre interferometers such as the SMA and ALMA.Comment: MNRAS Accepted 2009 July 2. 18 pages, 25 figures and 6 table

Brown representability for space-valued functors

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors.Comment: 19 pages, final version, accepted by the Israel Journal of Mathematic

Rotation Measure Synthesis of Galactic Polarized Emission with the DRAO 26-m Telescope

Radio polarimetry at decimetre wavelengths is the principal source of information on the Galactic magnetic field. The diffuse polarized emission is strongly influenced by Faraday rotation in the magneto-ionic medium and rotation measure is the prime quantity of interest, implying that all Stokes parameters must be measured over wide frequency bands with many frequency channels. The DRAO 26-m Telescope has been equipped with a wideband feed, a polarization transducer to deliver both hands of circular polarization, and a receiver, all operating from 1277 to 1762 MHz. Half-power beamwidth is between 40 and 30 arcminutes. A digital FPGA spectrometer, based on commercially available components, produces all Stokes parameters in 2048 frequency channels over a 485-MHz bandwidth. Signals are digitized to 8 bits and a Fast Fourier Transform is applied to each data stream. Stokes parameters are then generated in each frequency channel. This instrument is in use at DRAO for a Northern sky polarization survey. Observations consist of scans up and down the Meridian at a drive rate of 0.9 degree per minute to give complete coverage of the sky between declinations -30 degree and 90 degree. This paper presents a complete description of the receiver and data acquisition system. Only a small fraction of the frequency band of operation is allocated for radio astronomy, and about 20 percent of the data are lost to interference. The first 8 percent of data from the survey are used for a proof-of-concept study, which has led to the first application of Rotation Measure Synthesis to the diffuse Galactic emission obtained with a single-antenna telescope. We find rotation measure values for the diffuse emission as high as approximately 100 rad per square metre, much higher than recorded in earlier work.Comment: Accepted for publication in The Astronomical Journa

Smash products for secondary homotopy groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. In particular we prove a formula for the crossed effect of the cup-one product operation on unstable homotopy groups of spheres which was claimed by Barratt-Jones-Mahowald.Comment: We give a clearer description of the tensor product of symmetric sequences of quadratic pair module

The homotopy theory of dg-categories and derived Morita theory

The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the dg-category of $(C,D)$-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the \emph{classifying space of dg-categories} (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect) on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm. 8.15 is new. Minor corrections. Final version, to appear in Inventione