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In this chapter, we introduce the notion of pointed model categories and show that the homotopy category of a pointed model category has a suspension functor with an adjoint called the loop functor. This suspension functor is a generalisation of the standard notion of (reduced) suspension of pointed topological spaces. We shall also see that, in the case of chain complexes over a ring, this suspension functor is modelled by the shift functor. With these constructions in place, we can define the notion of a stable model category. The suspension and loop functors allow us to define cofibre and fibre sequences in an arbitrary pointed model category. These sequences are a generalisation of cofibre and fibre sequences for pointed spaces category of a pointed model category and are a useful aid to calculations. When the model category is also stable, these cofibre and fibre sequences form the basis of important additional structure on the homotopy category.
The aim of this chapter is to state the definition of a triangulated category and show that the homotopy category of a stable model category is a triangulated category. Triangulated categories were developed to axiomatise the structure of the derived category of an abelian category. This structure comes in the form of exact triangles, a replacement for the short exact sequences of an abelian category. The exact triangles of a stable model category are defined in terms of cofibre sequences. We then investigate the consequences for the homotopy category of a stable model category such as the agreement between fibre and cofibre sequences. Next, we introduce exact functors, which are functors compatible with the structures of triangulated categories. We will show that a Quillen functor of stable model categories induces an exact functor of the respective homotopy categories. We end the chapter with two overview sections. This first introduces the concept of Toda brackets and applies the theory to the stable homotopy category. The second gives an example of a triangulated category that does not arise from a stable model category.
In the crudest sense, stable homotopy theory is the study of those homotopy invariant constructions of spaces which are preserved by suspension. In this chapter, we show how there are naturally occurring situations which exhibit stable behaviour. We will discuss several historic attempts at constructing a “stable homotopy category” where this stable behaviour can be studied, and we relate these to the more developed notions of spectra and the Bousfield–Friedlander model structure. Of course, if one only wants to perform calculations of stable homotopy groups, to have certain spectral sequences or similar, then one does not need much of the formalism of model categories of spectra. But as soon as one wishes to move away from those tasks and consider other stable homotopy theories (such as G–equivariant stable homotopy theory for some group G) or to make serious use of a symmetric monoidal smash product in the context of “Brave New Algebra”, then the advantages of the more formal setup become overwhelming.
In this chapter, we introduce symmetric spectra and orthogonal spectra along with their associated stable model structures. These versions of spectra have various technical advantages over sequential spectra. Furthermore, they are Quillen equivalent to the category of sequential spectra (equipped with its stable model structure). Hence, one may choose between these models according to their relative strengths. The primary advantage of symmetric and orthogonal spectra is that these model categories are symmetric monoidal models for the stable homotopy category. We will examine these monoidal structures further later on and show that symmetric spectra and orthogonal spectra are monoidally Quillen equivalent. Several other models of spectra also exist, and we will give short introductions to these later in this chapter. We end the chapter with a result that, roughly speaking, says that any model for the stable homotopy category will be Quillen equivalent to sequential spectra.
Bousfield localisation, or more specifically, left Bousfield localisation, is an established tool to formally add more weak equivalences to a model category. The most common setting is localisation of spaces or spectra with respect to a homology theory: rather than the weak equivalences being isomorphisms of homotopy groups, one constructs a model structure with the homology isomorphisms as the weak equivalences. As a consequence, the homology isomorphisms become strict isomorphisms in the corresponding homotopy category. Therefore, we can think of Bousfield localisation as a good formal framework for inverting maps in the homotopy category. Typically, information is lost in this process, but some specific aspects may stand out clearer after localisation. We will see an example of this behaviour in the final section when we show that the p-local stable homotopy category has vast computational advantages over working with the stable homotopy category itself. We will also see how Bousfield localisation can help us gain insight into the deeper structure of the stable homotopy category via p-localisation, p-completion, K-theory and chromatic homotopy theory.
The aim of this chapter is to investigate symmetric monoidal products on our categories of spectra and the stable homotopy category. After motivating this monoidal product in terms of the smash product on spaces and the Spanier–Whitehead category, we show that symmetric spectra and orthogonal spectra are symmetric monoidal model categories. As a consequence, the stable homotopy category is a closed symmetric monoidal category, and this monoidal structure is compatible with the triangulated structure. Using this monoidal product, we can give a modern interpretation of Spanier–Whitehead duality and discuss model categories of ring spectra, modules over ring spectra and commutative ring spectra. We end the chapter with an overview of some of the fundamental properties of spectra and the stable homotopy category, demonstrating that they are central to the study of stable homotopy theory. First, we show that the positive stable model structure on symmetric spectra is initial amongst stable simplicial monoidal model categories. Second, we show that the homotopy category of any stable model category has an “action” of the stable homotopy category.
Bousfield and Friedlander defined the stable homotopy category in terms of the homotopy category of a model category of spectra. We will construct this model category following an approach similar to Mandell-May-Schwede-Shipley based on sequential spectra. A sequential spectrum is a sequence of pointed topological spaces (and structure maps), thus, a natural candidate for an analogue of weak homotopy equivalences are those maps of spectra inducing a weak homotopy equivalence at every level. However, we will see that these levelwise weak homotopy equivalences are not sufficient to define a class of weak equivalences leading to a meaningful stable homotopy theory. A key ingredient is the definition of homotopy groups of spectra and their isomorphisms. This generalises the notion of stable homotopy groups of topological spaces that we encountered earlier. Making these isomorphisms the weak equivalences of sequential spectra will give us a construction of our desired stable homotopy category. We end the chapter with an introduction to the Steenrod algebra and the Adams spectral sequence.
The beginning graduate student in homotopy theory is confronted with a vast literature on spectra that is scattered across books, articles and decades. There is much folklore but very few easy entry points. This comprehensive introduction to stable homotopy theory changes that. It presents the foundations of the subject together in one place for the first time, from the motivating phenomena to the modern theory, at a level suitable for those with only a first course in algebraic topology. Starting from stable homotopy groups and (co)homology theories, the authors study the most important categories of spectra and the stable homotopy category, before moving on to computational aspects and more advanced topics such as monoidal structures, localisations and chromatic homotopy theory. The appendix containing essential facts on model categories, the numerous examples and the suggestions for further reading make this a friendly introduction to an often daunting subject.
In this study we sought to identify profiles of talk during Head Start preschool mealtime conversations involving teachers and students. Videos of 44 Head Start classrooms’ lunch interactions were analyzed for the ratio of teacher–child talk and amount of academic vocabulary, and then coded for instances of academic/food, social/personal, and management talk to highlight the degree of hybridity of talk within this unique setting. Cluster analysis revealed four distinct patterns of teacher–child mealtime interactions in 44 Head Start preschool classrooms: classroom discourse, home discourse, hybrid-low, and hybrid-high. Multilevel models further demonstrated a relationship among these clusters of teacher–child interactions and children's end-of-year expressive vocabulary scores controlling for ratio of teacher–child talk and pre-test scores. Children in classrooms displaying a hybrid style of mealtime discourse made the greatest gains on measures of expressive vocabulary in contrast to their peers in classrooms displaying other discourse styles.
OBJECTIVES/SPECIFIC AIMS: Chlamydia trachomatis (CT) infection can lead to reproductive morbidity in women. Animal models suggest that protection against CT is mediated through the cytokine interferon-gamma (IFN-γ), produced by CD4+ T-cells, which clears CT through intracellular tryptophan depletion. In humans, correlates of protection remain to be elucidated, which hinders chlamydia vaccine development. Natural clearance of CT infection (e.g., clearance before antibiotics) may be an immunological correlate of protection, evidenced by (1) CT clearance without antibiotics; and (2) a 4-fold reduced risk of CT reinfection within 6 months. We have identified women with and without natural clearance of CT infection. By comparing these two groups of women, the role of IFN-γ-mediated natural clearance of CT infection will be investigated. METHODS/STUDY POPULATION: Through collaboration with a cohort study of CT-infected women, we have access to stored specimens from women who naturally cleared CT or had persisting CT infection. Using peripheral blood mononuclear cell (PBMC), we will assess whether natural clearance of CT infection is associated with IFN-γ-producing CD4+ T-cells by stimulating PBMC ex vivo with CT antigens using intracellular cytokine staining. We will also use cervicovaginal lavage (CVL) and untargeted High-Performance Liquid Chromatography-Mass Spectrometry to assess for tryptophan-dependent and -independent metabolic pathways associated with natural clearance of CT infection. RESULTS/ANTICIPATED RESULTS:: To date, IFN-γ has been measured in 10 women who did not clear CT infection, demonstrating that <20% of these women produced significant levels of IFN-γ. Women who naturally cleared CT have yet to be studied. Untargeted HPLC-MS has been performed on 6 women (3 who cleared matched to 3 with persisting CT infection). To date, 11 pathways that are significantly associated with natural clearance have been identified. DISCUSSION/SIGNIFICANCE OF IMPACT: The outcome of natural clearance of CT infection is distinct from women with persisting chlamydia. These studies may inform whether IFN-γ, produced by CD4+ T-cells, or tryptophan-dependent or -independent metabolic pathways are associated with natural clearance, which may advance chlamydia vaccine development.
Movement disorders associated with exposure to antipsychotic drugs are common and stigmatising but underdiagnosed.
To develop and evaluate a new clinical procedure, the ScanMove instrument, for the screening of antipsychotic-associated movement disorders for use by mental health nurses.
Item selection and content validity assessment for the ScanMove instrument were conducted by a panel of neurologists, psychiatrists and a mental health nurse, who operationalised a 31-item screening procedure. Interrater reliability was measured on ratings for 30 patients with psychosis from ten mental health nurses evaluating video recordings of the procedure. Criterion and concurrent validity were tested comparing the ScanMove instrument-based rating of 13 mental health nurses for 635 community patients from mental health services with diagnostic judgement of a movement disorder neurologist based on the ScanMove instrument and a reference procedure comprising a selection of commonly used rating scales.
Interreliability analysis showed no systematic difference between raters in their prediction of any antipsychotic-associated movement disorders category. On criterion validity testing, the ScanMove instrument showed good sensitivity for parkinsonism (90%) and hyperkinesia (89%), but not for akathisia (38%), whereas specificity was low for parkinsonism and hyperkinesia, and moderate for akathisia.
The ScanMove instrument demonstrated good feasibility and interrater reliability, and acceptable sensitivity as a mental health nurse-administered screening tool for parkinsonism and hyperkinesia.
SNP in the vitamin D receptor (VDR) gene is associated with risk of lower respiratory infections. The influence of genetic variation in the vitamin D pathway resulting in susceptibility to upper respiratory infections (URI) has not been investigated. We evaluated the influence of thirty-three SNP in eleven vitamin D pathway genes (DBP, DHCR7, RXRA, CYP2R1, CYP27B1, CYP24A1, CYP3A4, CYP27A1, LRP2, CUBN and VDR) resulting in URI risk in 725 adults in London, UK, using an additive model with adjustment for potential confounders and correction for multiple comparisons. Significant associations in this cohort were investigated in a validation cohort of 737 children in Manchester, UK. In all, three SNP in VDR (rs4334089, rs11568820 and rs7970314) and one SNP in CYP3A4 (rs2740574) were associated with risk of URI in the discovery cohort after adjusting for potential confounders and correcting for multiple comparisons (adjusted incidence rate ratio per additional minor allele ≥1·15, Pfor trend ≤0·030). This association was replicated for rs4334089 in the validation cohort (Pfor trend=0·048) but not for rs11568820, rs7970314 or rs2740574. Carriage of the minor allele of the rs4334089 SNP in VDR was associated with increased susceptibility to URI in children and adult cohorts in the United Kingdom.
In the last decade observations have been able to probe the evolution of the galaxy luminosity function, in particular showing a variation of its faint-end with redshift. We employ the data of the Cluster-EAGLE project, a set of cosmological, hydrodynamical zoom-in simulations of 30 galaxy clusters, to study the evolution of the galaxy luminostity functions in clusters with redshift. We compile a catalogue of simulated galaxies’ luminosities in the SDSS bands using the E-MILES spectra database, and taking into account dust attenuation. Stacked luminosity functions present little evolution with redshift of the faint-end slope from z=3.5 to z=0, regardless of the cluster mass.