4,894 research outputs found
Kan extensions and the calculus of modules for -categories
Various models of -categories, including quasi-categories,
complete Segal spaces, Segal categories, and naturally marked simplicial sets
can be considered as the objects of an -cosmos. In a generic
-cosmos, whose objects we call -categories, we introduce
modules (also called profunctors or correspondences) between
-categories, incarnated as as spans of suitably-defined fibrations with
groupoidal fibers. As the name suggests, a module from to is an
-category equipped with a left action of and a right action of ,
in a suitable sense. Applying the fibrational form of the Yoneda lemma, we
develop a general calculus of modules, proving that they naturally assemble
into a multicategory-like structure called a virtual equipment, which is known
to be a robust setting in which to develop formal category theory. Using the
calculus of modules, it is straightforward to define and study pointwise Kan
extensions, which we relate, in the case of cartesian closed -cosmoi,
to limits and colimits of diagrams valued in an -category, as
introduced in previous work.Comment: 84 pages; a sequel to arXiv:1506.05500; v2. new results added, axiom
circularity removed; v3. final journal version to appear in Alg. Geom. To
Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions
Consider a diagram of quasi-categories that admit and functors that preserve
limits or colimits of a fixed shape. We show that any weighted limit whose
weight is a projective cofibrant simplicial functor is again a quasi-category
admitting these (co)limits and that they are preserved by the functors in the
limit cone. In particular, the Bousfield-Kan homotopy limit of a diagram of
quasi-categories admit any limits or colimits existing in and preserved by the
functors in that diagram. In previous work, we demonstrated that the
quasi-category of algebras for a homotopy coherent monad could be described as
a weighted limit with projective cofibrant weight, so these results immediately
provide us with important (co)completeness results for quasi-categories of
algebras. These generalise most of the classical categorical results, except
for a well known theorem which shows that limits lift to the category of
algebras for any monad, regardless of whether its functor part preserves those
limits. The second half of this paper establishes this more general result in
the quasi-categorical setting: showing that the monadic forgetful functor of
the quasi-category of algebras for a homotopy coherent monad creates all limits
that exist in the base quasi-category, without further assumption on the monad.
This proof relies upon a more delicate and explicit analysis of the particular
weight used to define quasi-categories of algebras.Comment: 33 pages; a sequel to arXiv:1306.5144 and arXiv:1310.8279; v3: final
journal version with updated internal references to the new version of
"Homotopy coherent adjunctions and the formal theory of monads
Need, Merit or Self-Interest - What Determines the Allocation of Aid?
Previous studies into aid allocation have concluded that foreign aid is allocated not only according to development needs but also according to donor self-interest. We revisit this topic and allow for donor as well as recipient specific effects in our analysis. Our results indicate that roughly half of the predicted value of aid is determined by donor specific effects. Of the remaining variation, recipient need accounts for 36 percent and donor selfinterest or about 16 percent. This suggests that the previous literature has overstated the importance of donor self-interest. However, bilateral donors seem to place little importance on recipient merit. Recipient merit, measured by growth, democracy and human rights, accounts for only two percent of predicted aid.
Accidental immersion and unintentional drowning of rural children: An investigation for the Child Accident Prevention Foundation of New Zealand
In New Zealand drowning amongst preschoolers is one of the leading causes of death. The monetary and emotional costs to society are devastating and cannot be underestimated . The need to reduce the high number of deaths and the monetary and emotional costs prompted this research. This research presents the possibility for proactive measures to be taken in this area. In addition, it provides insightful knowledge for parents and educators alike. Ultimately, it seeks to reduce the number of child drownings that occur in the rural environment
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