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

Phantom Maps and Homology Theories

We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X,Y) of phantom maps X -> Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X,HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis's axiomatisation conjecture to an extension problem. We show that a certain functor S -> A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.Comment: 25 pages, AMSLaTeX, to appear in Topolog

Mitochondrial Na+ controls oxidative phosphorylation and hypoxic redox signalling

All metazoans depend on O2 delivery and consumption by the mitochondrial oxidative phosphorylation (OXPHOS) system to produce energy. A decrease in O2 availability (hypoxia) leads to profound metabolic rewiring. In addition, OXPHOS uses O2 to produce reactive oxygen species (ROS) that can drive cell adaptations through redox signalling, but also trigger cell damage1–4, and both phenomena occur in hypoxia4–8. However, the precise mechanism by which acute hypoxia triggers mitochondrial ROS production is still unknown. Ca2+ is one of the best known examples of an ion acting as a second messenger9, yet the role ascribed to Na+ is to serve as a mere mediator of membrane potential and collaborating in ion transport10. Here we show that Na+ acts as a second messenger regulating OXPHOS function and ROS production by modulating fluidity of the inner mitochondrial membrane (IMM). We found that a conformational shift in mitochondrial complex I during acute hypoxia11 drives the acidification of the matrix and solubilization of calcium phosphate precipitates. The concomitant increase in matrix free-Ca2+ activates the mitochondrial Na+/Ca2+ exchanger (NCLX), which imports Na+ into the matrix. Na+ interacts with phospholipids reducing IMM fluidity and mobility of free ubiquinone between complex II and complex III, but not inside supercomplexes. As a consequence, superoxide is produced at complex III, generating a redox signal. Inhibition of mitochondrial Na+ import through NCLX is sufficient to block this pathway, preventing adaptation to hypoxia. These results reveal that Na+ import into the mitochondrial matrix controls OXPHOS function and redox signalling through an unexpected interaction with phospholipids, with profound consequences in cellular metabolism

MOOCs in the Language Classroom: Using MOOCs as Complementary Materials to Support Self-Regulated Language Learning

In this chapter, the author discusses the role of Massive Open Online Courses (MOOCs) in the field of foreign language education. It examines the relationship between MOOCs and self-regulated learning, and goes on to explore the repurposing of content-based MOOCs as complementary materials to foster students' self-regulated behaviour and practice of the target language inside and out the language classroom. A research project that has explored self-regulated language learning in a blended context with content-based MOOCs is also discussed here, with particular reference to the self-regulatory strategies employed by a group of language students at University level. A series of recommendations that supports the use of content-based MOOCs as part of face-to-face language courses are provided at the end of this chapter