Complete Boolean algebras are Bousfield lattices

Given a complete Heyting algebra we construct an algebraic tensor triangulated category whose Bousfield lattice is the Booleanization of the given Heyting algebra. As a consequence we deduce that any complete Boolean algebra is the Bousfield lattice of some tensor triangulated category. Using the same ideas we then give two further examples illustrating some interesting behaviour of the Bousfield lattice.Comment: 10 pages, update to clarify the products occurring in the main constructio

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories

The Dold-Kan Correspondence and Coalgebra Structures

By using the Dold-Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected coalgebras and prove a Quillen equivalence between them.Comment: 24 pages. Accepted by the Journal of Homotopy and Related Structures. Online 28 November 201

Understanding the small object argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.Comment: 42 pages; supersedes the earlier arXiv preprint math/0702290; v2: final journal version, minor corrections onl

Rigidity and exotic models for v1-local G-equivariant stable homotopy theory

We prove that the v1-local G-equivariant stable homotopy category for G a finite group has a unique G-equivariant model at p=2. This means that at the prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p≥5 equivariant rigidity fails in general

Anaerobic membrane bioreactors for municipal wastewater treatment : a literature review

Currently, there is growing scientific interest in the development of more economic, efficient and environmentally friendly municipal wastewater treatment technologies. Laboratory and pilot-scale surveys have revealed that the anaerobic membrane bioreactor (AnMBR) is a promising alternative for municipal wastewater treatment. Anaerobic membrane bioreactor technology combines the advantages of anaerobic processes and membrane technology. Membranes retain colloidal and suspended solids and provide complete solid−liquid separation. The slow-growing anaerobic microorganisms in the bioreactor degrade the soluble organic matter, producing biogas. The low amount of produced sludge and the production of biogas makes AnMBRs favorable over conventional biological treatment technologies. However, the AnMBR is not yet fully mature and challenging issues remain. This work focuses on fundamental aspects of AnMBRs in the treatment of municipal wastewater. The important parameters for AnMBR operation, such as pH, temperature, alkalinity, volatile fatty acids, organic loading rate, hydraulic retention time and solids retention time, are discussed. Moreover, through a comprehensive literature survey of recent applications from 2009 to 2021, the current state of AnMBR technology is assessed and its limitations are highlighted. Finally, the need for further laboratory, pilot- and full-scale research is addressed