Enhanced 2-categories and limits for lax morphisms

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is "enhanced", by specifying which of these weak morphisms are actually strict, can be thought of as category enriched over a particular base cartesian closed category F. We give a complete characterization, in terms of F-enriched category theory, of the limits which exist in such 2-categories of categories with extra structure.Comment: 77 pages; v2 minor changes only, to appear in Advance

Idempotent splittings, colimit completion, and weak aspects of the theory of monads

We show that some recent constructions in the literature, named `weak' generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.Comment: LaTeX source 23 pages; v3 final version, minor corrections only, to appear in Journal of Pure and Applied Algebr

Weak bimonads and weak Hopf monads

We define a weak bimonad as a monad T on a monoidal category M with the property that the Eilenberg-Moore category M^T is monoidal and the forgetful functor from M^T to M is separable Frobenius. Whenever M is also Cauchy complete, a simple set of axioms is provided, that characterizes the monoidal structure of M^T as a weak lifting of the monoidal structure of M . The relation to bimonads, and the relation to weak bimonoids in a braided monoidal category are revealed. We also discuss antipodes, obtaining the notion of weak Hopf monad.Comment: 29 pages; version 2 minor corrections and added references, also added remark 4.3; title changed from "Weak bimonads" to "Weak bimonads and weak Hopf monads"; to appear in Journal of Algebr

Introduction:Diagnostics, medical testing, and value in medical anthropology

Introduction to the Special Issue on Diagnostics, Medical testing, and Valu

Lex colimits

Many kinds of categorical structure require the existence of finite limits, of colimits of some specified type, and of "exactness" conditions between the finite limits and the specified colimits. Some examples are the notions of regular, or Barr-exact, or lextensive, or coherent, or adhesive category. We introduce a general notion of exactness, of which each of the structures listed above, and others besides, are particular instances. The notion can be understood as a form of cocompleteness "in the lex world" -- more precisely, in the 2-category of finitely complete categories and finite-limit preserving functors.Comment: 38 pages; v2: final journal version, various minor changes and new Section 5.9 dealing with filtered colimit

A Quillen model structure for Gray-categories

A Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.Comment: v2: fuller discussion of relationship with work of Berger; localizations are done directly with simplicial set

Urinary Phthalate Metabolites and Biomarkers of Oxidative Stress in a Mexican-American Cohort: Variability in Early and Late Pregnancy.

People are exposed to phthalates through their wide use as plasticizers and in personal care products. Many phthalates are endocrine disruptors and have been associated with adverse health outcomes. However, knowledge gaps exist in understanding the molecular mechanisms associated with the effects of exposure in early and late pregnancy. In this study, we examined the relationship of eleven urinary phthalate metabolites with isoprostane, an established marker of oxidative stress, among pregnant Mexican-American women from an agricultural cohort. Isoprostane levels were on average 20% higher at 26 weeks than at 13 weeks of pregnancy. Urinary phthalate metabolite concentrations suggested relatively consistent phthalate exposures over pregnancy. The relationship between phthalate metabolite concentrations and isoprostane levels was significant for the sum of di-2-ethylhexyl phthalate and the sum of high molecular weight metabolites with the exception of monobenzyl phthalate, which was not associated with oxidative stress at either time point. In contrast, low molecular weight metabolite concentrations were not associated with isoprostane at 13 weeks, but this relationship became stronger later in pregnancy (p-value = 0.009 for the sum of low molecular weight metabolites). Our findings suggest that prenatal exposure to phthalates may influence oxidative stress, which is consistent with their relationship with obesity and other adverse health outcomes

Coalgebraic Semantics for Timed Processes

We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad ” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate. Key words: time domains, timed transition systems, evolution comonads, delay operators, structural operational semantics, modularity, distributive laws