We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.Comment: 35 pages; major changes in section 5, minor changes elsewher

volV) (0123456789().,-volV) can move into the groundwater specified in the exposure assessment option as well as the magnitude of residues in groundwater. The objective may also include determining degradation rates in soil as a function of depth, persistence and movement of residues in groundwater, efficacy of mitigation measures, or confirmation of more detailed studies on a wider range of sites. Sampling schedules should consider the expected time required for an active substance to move through the soil into groundwater, as well as expected persistence in both soil and groundwater. Movement and persistence can be affected by both site characteristics and properties of the active substance and its metabolites. The need to tailor study designs to objectives, exposure assessment options, compound properties and site characteristics complicates the development of standardised study designs. Therefore, this report includes a number of example designs.Other key points that must be addressed by study designs are the vulnerability of the chosen sites compared to the vulnerability of all use areas supported by the study, the product use before and during the study, and the connectivity of the sampled groundwater to treated fields. Demonstrating connectivity (a quality criterion in the EU assessment of monitoring sites to exclude false negative measurements) is more challenging for catchment or aquifer monitoring compared to shallow wells installed as part of in-field or edge-of-field studies. This report includes an extensive discussion on assessing vulnerability of monitoring sites. This includes information on different approaches to vulnerability assessment and mapping as well as for setting monitoring sites into context. Lists of available methods and data sources available at the European level are also included. In addition to information on study design and estimating vulnerability, this report includes information on a number of other topics: avoiding contamination during sampling and/or analysis, avoiding influencing residue movement as a result of purging during sampling, and proper study documentation (Good Laboratory Practices and/or quality criteria). Procedures that are discussed include site selection (new or existing wells), installation of monitoring wells, sample collection, and analysis of samples. The report also provides information on causes of outliers (abnormally high concentrations not the result of normal leaching through soil), the use of public monitoring data, information on further hydrological characterisation (such as use of tracers, groundwater age dating, and geophysical methods), and information that should be included in reports providing results of groundwater studies. AbstractGroundwater monitoring is recommended as a higher-tier option in the regulatory groundwater assessment of crop protection products in the European Union. However, to date little guidance has been provided on the study designs. The SETAC EMAG-Pest GW group (a mixture of regulatory, academic, and industry scien...

In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones. Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.Comment: Revised version; 32 page

In a semi-abelian context, we study the condition (NH) asking that Higgins commutators of normal subobjects are normal subobjects. We provide examples of categories that do or do not satisfy this property. We focus on the relationship with the "Smith is Huq" condition (SH) and characterise those semi-abelian categories in which both (NH) and (SH) hold in terms of reflection and preservation properties of the change of base functors of the fibration of points.Comment: 15 pages; final published versio