Stems and Spectral Sequences

We introduce the category Pstem[n] of n-stems, with a functor P[n] from spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of a space. We show how to associate to each simplicial n-stem Q an (n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem of a simplicial space X, the truncated spectral sequence for Q is the truncation of the usual homotopy spectral sequence of X. Similar results are also proven for cosimplicial n-stems. They are helpful for computations, since n-stems in low degrees have good algebraic models

Segal-type algebraic models of n-types

For each n\geq 1 we introduce two new Segal-type models of n-types of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up to homotopy by such models via an explicit algebraic fundamental n-fold groupoid functor. We compare these models to Tamsamani's weak n-groupoids, and extract from them a model for (k-1)connected n-typesComment: Added index of terminology and notation. Minor amendments and added details is some definitions and proofs. Some typos correcte

Mapping spaces in Quasi-categories

We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way

Rigidification of quasi-categories

We give a new construction for rigidifying a quasi-category into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an application of these methods, we use our model to reprove some basic facts from Lurie's "Higher Topos Theory" regarding the rigidification process.Comment: 26 page

Model Categories for Orthogonal Calculus

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.Comment: 36 pages, added a new section introducing spaces with a group action, minor corrections from previous versio

FGDB: Database of Follicle Stimulating Hormone Glycans

Glycomics, the study of the entire complement of sugars of an organism has received significant attention in the recent past due to the advances made in high throughput mass spectrometry technologies. These analytical advancements have facilitated the characterization of glycans associated with the follicle-stimulating hormones (FSH), which play a central role in the human reproductive system both in males and females utilizing regulating gonadal (testicular and ovarian) functions. The irregularities in FSH activity are also directly linked with osteoporosis. The glycoanalytical studies have been tremendously helpful in understanding the biological roles of FSH. Subsequently, the increasing number of characterized FSH glycan structures and related glycoform data has thrown a challenge to the glycoinformatics community in terms of data organization, storage and access. Also, a user-friendly platform is needed for providing easy access to the database and performing integrated analysis using a high volume of experimental data to accelerate FSH-focused research. FSH Glycans DataBase (FGDB) serves as a comprehensive and unique repository of structures, features, and related information of glycans associated with FSH. Apart from providing multiple search options, the database also facilitates an integrated user-friendly interface to perform the glycan abundance and comparative analyses using experimental data. The automated integrated pipelines present the possible structures of glycans and variants of FSH based on the input data, and allow the user to perform various analyses. The potential application of FGDB will significantly help both glycoinformaticians as well as wet-lab researchers to stimulate the research in this area. FGDB web access: https://fgdb.unmc.edu/

On realizing diagrams of Pi-algebras

Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Pi-algebras. This extends a program begun in [J. Pure Appl. Alg. 103 (1995) 167-188] and [Topology 43 (2004) 857-892] to study the realization of a single Pi-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations.Comment: This is the version published by Algebraic & Geometric Topology on 21 June 200

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

Carbon payments can cost-effectively improve logging sustainability in the Amazon

Selective logging is pervasive across the tropics and unsustainable logging depletes forest biodiversity and carbon stocks. Improving the sustainability of logging will be crucial for meeting climate targets. Carbon-based payment for ecosystem service schemes, including REDD+, give economic value to standing forests and can protect them from degradation, but only if the revenue from carbon payments is greater than the opportunity cost of forgone or reduced logging. We currently lack understanding of whether carbon payments are feasible for protecting Amazonian forests from logging, despite the Amazon holding the largest unexploited timber reserves and an expanding logging sector. Using financial data and inventories of >660,000 trees covering 52,000 ha of Brazilian forest concessions, we estimate the carbon price required to protect forests from logging. We estimate that a carbon price of $7.90 per tCO2 is sufficient to match the opportunity costs of all logging and fund protection of primary forest. Alternatively, improving the sustainability of logging operations by ensuring a greater proportion of trees are left uncut requires only slightly higher investments of$7.97–10.45 per tCO2. These prices fall well below the current compliance market rate and demonstrate a cost-effective opportunity to safeguard large tracts of the Amazon rainforest from further degradation

Formes differentielles generalisees sur une operade et modeles algebriques des fibrations

On construit des foncteurs de formes differentielles generalisees. Ceux-ci, dans le cas d'espaces nilpotents de type fini, determinent le type d'homotopie faible des espaces. Ils sont munis, d'une maniere elementaire et naturelle, de l'action de cup-i produits. Pour les algebres commutatives a homotopit pres (algebres sur une resolution cofibrante de l'operade des algebres commutatives), on demontre en utilisant les formes differentielles generalisees que le modele de la fibre d'une application simpliciale est la cofibre du modele de ce morphisme. We construct functors of generalized differential forms. In the case of nilpotent spaces of finite type, they determine the weak homotopy type of the spaces. Moreover they are equipped, in an elementary and natural way, with the action of cup-i products. Working with commutative algebras up to homotopy (viewed as algebras over a cofibrant resolution of the operad of commutative algebras), we show using these functors that the model of the fiber of a simplicial map is the cofiber of the algebraic model of this map.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-5.abs.htm