Displayed Categories

We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.Comment: v3: Revised and slightly expanded for publication in LMCS. Theorem numbering change

Displayed Categories

We introduce and develop the notion of displayed categories. A displayed category over a category C is equivalent to "a category D and functor F : D -> C", but instead of having a single collection of "objects of D" with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments

Displayed Monoidal Categories for the Semantics of Linear Logic

We present a formalization of different categorical structures used to interpret linear logic. Our formalization takes place in UniMath, a library of univalent mathematics based on the Coq proof assistant.All the categorical structures we formalize are based on monoidal categories. As such, one of our contributions is a practical, usable library of formalized results on monoidal categories. Monoidal categories carry a lot of structure, and instances of monoidal categories are often built from complicated mathematical objects. This can cause challenges of scalability, regarding both the vast amount of data to be managed by the user of the library, as well as the time the proof assistant spends on checking code. To enable scalability, and to avoid duplication of computer code in the formalization, we develop "displayed monoidal categories". These gadgets allow for the modular construction of complicated monoidal categories by building them in layers; we demonstrate their use in many examples. Specifically, we define linear-non-linear categories and construct instances of them via Lafont categories and linear categories

Focus markers that link topic and comment

This talk deals with the interdependence between the pragmatic categories topic and focus as displayed by certain alleged focus marking particles of some West African languages

Ohio's statewide land use inventory: An operational approach for applying LANDSAT data to state, regional and local planning programs

The programmatic, technical, user application, and cost factors associated with the development of an operational, statewide land use inventory from LANDSAT data are described. The LANDSAT multispectral data are subjected to geometrical and categorical processing to produce map files for each of the 200 fifteen (15) minute quads covering Ohio. Computer compatible tapes are rescanned to produce inventory tapes which identify eight (8) Level I land use categories and a variety of Level II categories. The inventory tapes are processed through a series of ten (10) software programs developed by the State of Ohio. The net result is a computerized inventory which can be displayed in map or tabular form for various geographic units, at a variety of scales and for selected categories of usage. The computerized inventory data files are applied to technical programs developed by the various state agencies to be used in state, regional, and local planning programs

Breakfast clubs: Starting the day in a positive way

Breakfast clubs are widely promoted as having a beneficial impact on children’s behavior at the start of the school day, which can be conducive to their learning within the classroom. However, the few available studies that have considered the impact of breakfast club attendance on children’s behavior have yielded mixed results and no studies to date have directly observed children’s behavior within the breakfast club setting. Using a combination of real-time observation and filmed breakfast club footage, the aims of the current study were to: (1) devise a set of observational criteria appropriate for use in the breakfast club setting; (2) investigate the occurrence of both positive and negative behaviors. A sample of 30 children aged between 3 and 11 years were recruited from 3, opportunistically sampled primary school breakfast clubs in the North East of England, UK. The behaviors they displayed within the breakfast club setting on two separate days were observed and coded for subsequent analysis. Results of the investigation showed that children’s behavior could be classified into three positive and three negative behavioral categories. Using these categories to code children’s behavior as they engaged in breakfast club showed that children displayed more positive than negative behaviors within the breakfast club setting and this was the case regardless of the type of activity (i.e., quiet or boisterous) children were involved in. Findings are discussed in relation to breakfast club policy, implementation, and evaluation

Associations between daily sitting time and the combinations of lifestyle risk factors in men

Background: Understanding the reciprocal role that multiple problematic behaviours play in men's health is important for intervention delivery and for reducing the healthcare burden. Data regarding the concurrence of problematic health behaviours is currently limited but offers insights into risk profiles, and should now include total time spent sitting/day. Methods: Self-reported data on lifestyle health behaviours was collected from 232 men aged ≥18 years who engaged in a men's health promotion programme delivered by 16 English Premier League Clubs. Results: Men at risk due to high sitting display multiple concurrent lifestyle risk factors, 88.6% displayed at least two ancillary risk factors and were three times more likely to report ≥2 lifestyle risk factors (OR. =3.13, 95% confidence interval (CI). =1.52-6.42) than those with low sitting risk. Significant differences in the mean number of risk factors reported between those participants in the higher risk (2.43. ±. 0.90) and lower risk (2.13. ±. 0.96) sitting categories were also found (P=0.015). Conclusions: Hard-to-reach men displayed multiple problematic concurrent behaviours, strongly linked to total sitting time. © 2012 WPMH GmbH

Measurement of inpatient care for small and sick newborns

A spreadsheet containing results of a global survey to classify 18 newborn care interventions. The survey was aimed at individuals working in maternal and newborn health and completed by 262 respondents located in 61 countries. Linking to emergency obstetric care levels, respondents were asked to rank the level of care they thought appropriate for health systems in Low & Middle Income Countries (LMICs) to provide based upon the following categories: [1] “routine care at birth”, [2] “special care”, [3] “intensive care”, or [4] “not appropriate for any level”. Results are displayed with a tab for each intervention and show variation in respondent characteristics with chi-squared value

Flight investigation of cockpit-displayed traffic information utilizing coded symbology in an advanced operational environment

Traffic symbology was encoded to provide additional information concerning the traffic, which was displayed on the pilot's electronic horizontal situation indicators (EHSI). A research airplane representing an advanced operational environment was used to assess the benefit of coded traffic symbology in a realistic work-load environment. Traffic scenarios, involving both conflict-free and conflict situations, were employed. Subjective pilot commentary was obtained through the use of a questionnaire and extensive pilot debriefings. These results grouped conveniently under two categories: display factors and task performance. A major item under the display factor category was the problem of display clutter. The primary contributors to clutter were the use of large map-scale factors, the use of traffic data blocks, and the presentation of more than a few airplanes. In terms of task performance, the cockpit-displayed traffic information was found to provide excellent overall situation awareness. Additionally, mile separation prescribed during these tests