Magnitude and magnitude homology of filtered sets enriched categories

In this article, we give a framework for studying the Euler characteristic and its categorification of objects across several areas of geometry, topology and combinatorics. That is, the magnitude theory of filtered sets enriched categories. It is a unification of the Euler characteristic of finite categories and it the magnitude of metric spaces, both of which are introduced by Leinster. Our definitions cover a class of metric spaces which is broader than the original ones, so that magnitude (co)weighting of infinite metric spaces can be considered. We give examples of the magnitude from various research areas containing the Poincar\'{e} polynomial of ranked posets and the growth function of finitely generated groups. In particular, the magnitude homology gives categorifications of them. We also discuss homotopy invariance of the magnitude homology and its variants. Such a homotopy includes digraph homotopy and r-closeness of Lipschitz maps. As a benefit of our categorical view point, we generalize the notion of Grothendieck fibrations of small categories to our enriched categories, whose restriction to metric spaces is a notion called metric fibration that is initially introduced by Leinster. It is remarkable that the magnitude of such a fibration is a product of those of the fiber and the base. We especially study fibrations of graphs, and give examples of graphs with the same magnitude but are not isomorphic.Comment: 35 pages, 1 figur

Functional Traits Co-Occurring with Mobile Genetic Elements in the Microbiome of the Atacama Desert

Mobile genetic elements (MGEs) play an essential role in bacterial adaptation and evolution. These elements are enriched within bacterial communities from extreme environments. However, very little is known if specific genes co-occur with MGEs in extreme environments and, if so, what their function is. We used shotgun-sequencing to analyse the metagenomes of 12 soil samples and characterized the composition of MGEs and the genes co-occurring with them. The samples ranged from less arid coastal sites to the inland hyperarid core of the Atacama Desert, as well as from sediments below boulders, protected from UV-irradiation. MGEs were enriched at the hyperarid sites compared with sediments from below boulders and less arid sites. MGEs were mostly co-occurring with genes belonging to the Cluster Orthologous Group (COG) categories “replication, recombination and repair,” “transcription” and “signal transduction mechanisms.” In general, genes coding for transcriptional regulators and histidine kinases were the most abundant genes proximal to MGEs. Genes involved in energy production were significantly enriched close to MGEs at the hyperarid sites. For example, dehydrogenases, reductases, hydrolases and chlorite dismutase and other enzymes linked to nitrogen metabolism such as nitrite- and nitro-reductase. Stress response genes, including genes involved in antimicrobial and heavy metal resistance genes, were rarely found near MGEs. The present study suggests that MGEs could play an essential role in the adaptation of the soil microbiome in hyperarid desert soils by the modulation of housekeeping genes such as those involved in energy production.EC/FP7/339231/EU/Habitability of Martian Environments: Exploring the Physiological and Environmental Limits of Life/HOM

Comparing the orthogonal and homotopy functor calculi

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor which sends a vector space V to F evaluated at the one-point compactification of V. In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.Comment: 28 pages, sequel to Capturing Goodwillie's Derivative, arXiv:1406.042

Gabriel-Ulmer duality for topoi and its relation with site presentations

Let $\kappa$ be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally $\kappa$-presentable categories with $\kappa$-accessible right adjoints to its locally full sub-2-category of $\kappa$-presentable Grothendieck topoi with geometric $\kappa$-accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of $\kappa$-small cocomplete categories with $\kappa$-colimit preserving functors arising as the corresponding 2-category of presentations via the restriction. We analyse the relation of these presentations of Grothendieck topoi with site presentations and we show that the 2-category of locally $\kappa$-presentable Grothendieck topoi with geometric $\kappa$-accessible morphisms is a reflective sub-bicategory of the full sub-2-category of the 2-category of sites with morphisms of sites genearated by the weakly $\kappa$-ary sites in the sense of Shulman [37].Comment: 25 page

Global orthogonal spectra

For any finite group G, there are several well-established definitions of a G-equivariant spectrum. In this paper, we develop the definition of a global orthogonal spectrum. Loosely speaking, this is a coherent choice of orthogonal G-spectrum for each finite group G. We use the framework of enriched indexed categories to make this precise. We also consider equivariant K-theory and Spin^c-cobordism from this perspective, and we show that the Atiyah--Bott--Shapiro orientation extends to the global context.Comment: 17 page

Tensor products of finitely cocomplete and abelian categories

The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given.Comment: 14 page