A type theory for synthetic ∞\infty-categories

We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary types. We define Segal types, in which binary composites exist uniquely up to homotopy; this automatically ensures composition is coherently associative and unital at all dimensions. We define Rezk types, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities - a "local univalence" condition. And we define covariant fibrations, which are type families varying functorially over a Segal type, and prove a "dependent Yoneda lemma" that can be viewed as a directed form of the usual elimination rule for identity types. We conclude by studying homotopically correct adjunctions between Segal types, and showing that for a functor between Rezk types to have an adjoint is a mere proposition. To make the bookkeeping in such proofs manageable, we use a three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which can be abstracted over using "extension types" that generalize the path-types of cubical type theory. In an appendix, we describe the motivating semantics in the Reedy model structure on bisimplicial sets, in which our Segal and Rezk types correspond to Segal spaces and complete Segal spaces.Comment: 78 pages; v2 has minor updates inspired by discussions at the Mathematics Research Community on Homotopy Type Theory; v3 incorporates many expository improvements suggested by the referee; v4 is the final journal version to appear in Higher Structures with a more precise syntax for our type theory with shape

The Mass of Dwarf Planet Eris

The discovery of dwarf planet Eris was followed shortly by the discovery of its satellite, Dysnomia, but the satellite orbit, and thus the system mass, was not known. New observations with the Keck Observatory and the Hubble Space Telescopes show that Dysnomia has a circular orbit with a radius of 37,350 ± 140 (1-σ) kilometers and a 15.774 ± 0.002 day orbital period around Eris. These orbital parameters agree with expectations for a satellite formed out of the orbiting debris left from a giant impact. The mass of Eris from these orbital parameters is 1.67 × 10^(22) ± 0.02 × 10^(22) kilograms, or 1.27 ± 0.02 that of Pluto

The Married Widow: Marriage Penalties Matter!

Marriage penalties are a controversial feature of many government policies. Empirical evidence of their behavioral effects is quite mixed. This is surprising because economic theory predicts that they should have an impact on the headship decision. We investigate the removal of marriage penalties from the surviving spouse pensions of the Canadian public pension system in the 1980s. These reforms provide a simple and transparent source of identification. Our results indicate that marriage penalties can have large and persistent effects on marriage decisions. We also present evidence suggesting that it is individuals with characteristics correlated with greater wealth who respond to the penalties.

Increasing Low-income Mothers’ Educational Attainment: Implications for Anti-poverty Programs and Policy

Context: Emerging research indicates parental educational attainment is not always stable over time, particularly among young adults with lower levels of income and educational attainment. Though increases in postsecondary education are often highlighted as a route to greater earnings among higher-income students, it is unclear whether increases in parental educational attainment can improve the socioeconomic circumstances of low-income families. Objective: The first goal of the current study was to determine whether low-income mothers increased their educational attainment over a 6-year period as their children transitioned from early childhood through elementary school. Second, the current study examined a range of individual characteristics that may help or hinder a mother’s re-entrance into education. Last, associations between increased maternal education and indicators of family socioeconomic resources were examined to determine ways that increased education among low-income mothers of young children may serve as a mechanism to reduce poverty or other poverty-related risks. Design and Sample: Data for this study come from the Chicago School Readiness Project (CSRP), a cluster randomized control trial of Head Start centers and a longitudinal follow-up of children and their families. The current study included 432 participants. Of those participants, 97% were the child’s mother or female caregiver, 70% lived below the Federal Poverty Line at baseline, and 93% identified as a racial/ethnic minority (i.e., African American, black, or Hispanic). Main Outcome Measures: Maternal educational attainment was collected at 4 time-points across a 6-year period. From these data, a binary variable was created to indicate whether (1) or not (0) mothers increased their educational attainment. Maternal report of household income, unemployment status, and poverty-related risk were examined as indicators of family socioeconomic resources. Results: Thirty-nine percent of mothers increased their educational attainment over the 6-year period of study, and the majority of those mothers attained additional degrees rather than years of schooling alone. Mothers whose children attended treatment-assigned preschool classrooms at baseline were subsequently more likely to increase their educational attainment over time than were mothers of children who initially attended control-assigned classrooms in preschool. Analyses of the roles of parental characteristics in predicting gains in maternal education suggest that mothers who reported greater depressive symptomatology were less likely to increase their educational attainment. Increases in educational attainment, in turn, were positively associated with income earned in subsequent years of our longitudinal follow-up study and negatively associated with maternal unemployment and poverty-related risk when children were in 5th grade. Conclusions: Increases in parent educational attainment were impressive for our sample of low-income mothers, given their exposure to a range of poverty-related risks. Furthermore, our analyses support prior research suggesting that increases in maternal educational attainment may serve as an important mechanism to reduce families’ experience of income poverty

Can switching fuels save water? A life cycle quantification of freshwater consumption for Texas coal-and natural gas-fired electricity

Thermal electricity generation is a major consumer of freshwater for cooling, fuel extraction and air emissions controls, but the life cycle water impacts of different fossil fuel cycles are not well understood. Much of the existing literature relies on decades-old estimates for water intensity, particularly regarding water consumed for fuel extraction. This work uses contemporary data from specific resource basins and power plants in Texas to evaluate water intensity at three major stages of coal and natural gas fuel cycles: fuel extraction, power plant cooling and power plant emissions controls. In particular, the water intensity of fuel extraction is quantified for Texas lignite, conventional natural gas and 11 unconventional natural gas basins in Texas, including major second-order impacts associated with multi-stage hydraulic fracturing. Despite the rise of this water-intensive natural gas extraction method, natural gas extraction appears to consume less freshwater than coal per unit of energy extracted in Texas because of the high water intensity of Texas lignite extraction. This work uses new resource basin and power plant level water intensity data to estimate the potential effects of coal to natural gas fuel switching in Texas’ power sector, a shift under consideration due to potential environmental benefits and very low natural gas prices. Replacing Texas’ coal-fired power plants with natural gas combined cycle plants (NGCCs) would reduce annual freshwater consumption in the state by an estimated 53 billion gallons per year, or 60% of Texas coal power’s water footprint, largely due to the higher efficiency of NGCCs.Mechanical Engineerin

Coalgebraic models for combinatorial model categories

We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant.Comment: 12 pages; v2: final journal version with minor improvements suggested by the refere

Computing Homology Invariants of Legendrian Knots

The Chekanov-Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov-Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This article gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence. First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| is less than or equal to 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff need not have isomorphic homology groups.Comment: 18 pages, 11 figure