79,243 research outputs found
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 âreplication, recombination and repair,â âtranscriptionâ and âsignal transduction mechanisms.â 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 be a regular cardinal. We study Gabriel-Ulmer duality when one
restricts the 2-category of locally -presentable categories with
-accessible right adjoints to its locally full sub-2-category of
-presentable Grothendieck topoi with geometric -accessible
morphisms. In particular, we provide a full understanding of the locally full
sub-2-category of the 2-category of -small cocomplete categories with
-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 -presentable Grothendieck topoi with
geometric -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 -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
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