Topology from enrichment: the curious case of partial metrics

For any small quantaloid \Q, there is a new quantaloid \D(\Q) of diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa), and then it is particularly interesting to compare categories enriched in \Q with categories enriched in \D(\Q). Taking Lawvere's quantale of extended positive real numbers as base quantale, \Q-categories are generalised metric spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-enriched category has a canonical closure operator on its set of objects: this makes for a functor from quantaloid-enriched categories to closure spaces. Under mild necessary-and-sufficient conditions on the base quantaloid, this functor lands in the category of topological spaces; and an involutive quantaloid is Cauchy-bilateral (a property discovered earlier in the context of distributive laws) if and only if the closure on any enriched category is identical to the closure on its symmetrisation. As this now applies to metric spaces and partial metric spaces alike, we demonstrate how these general categorical constructions produce the "correct" definitions of convergence and Cauchyness of sequences in generalised partial metric spaces. Finally we describe the Cauchy-completion, the Hausdorff contruction and exponentiability of a partial metric space, again by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a revised section on Cauchy sequences in a partial metric spac

New combinations in Lactifluus, 2 : L. subg. Gerardii

In this second of a series of three papers, new combinations in the genus Lactifluus are proposed. This paper treats Lactarius subg. Gerardii (proposed here as new combination in Lactifluus). In this subgenus 17 combinations at species level are proposed

Exponentiable functors between quantaloid-enriched categories

Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor's exponentiability follows from the works of Niefield [1980] and Dyckhoff and Tholen [1987].Comment: 10 pages; correction of flaw in proo

Towards Stone duality for topological theories

AbstractIn the context of categorical topology, more precisely that of T-categories (Hofmann, 2007 [8]), we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of “Stone duality”, and show that Cauchy completeness of a T-category is precisely its sobriety

\u3ci\u3eHypsugo stubbei\u3c/i\u3e sp. nov., A Novel Cryptic Bat Species of the Genus \u3ci\u3eHypsugo\u3c/i\u3e (Vespertilionidae, Chiroptera, Mammalia) from Mongolia

The occurrence of two members of the genus Hypsugo, namely H. alaschanicus and H. savii caucasicus, have been reported for Mongolia in the literature. Due to various taxonomic reassignments within and between genera, the number of records for the genus Hypsugo in Mongolia is quite scarce and sometimes not resolved at species or subspecies level. Despite recognition of the two above-mentioned species, recent reports based on genetic analyses describe only new and further records of H. alaschanicus. Thus, it exists a large uncertainty regarding the occurrence and distribution of H. savii caucasicus in Mongolia. Here, our efforts in gaining a deeper understanding towards the occurrence and distribution of Hypsugo species in Mongolia are described. A combination of genetic and morphological analyses of collected material from Hypsugo specimens revealed the existence of a genetically largely distant Hypsugo clade. Therefore, a new and cryptic Hypsugo species is proposed which is named after Prof. Dr. Michael Stubbe for his continuous, long-standing and significant contributions into the biological exploration of Mongolia. Hypsugo stubbei sp. nov. differs by at least 8.4 % and 9 % to the closest Western Palearctic distributed H. cf. darwinii and H. savii as well as at least 11.3 % to the Easter Palearctic (including Mongolia) distributed H. alaschanicus based on the first 798 nucleotides of the gene encoding the mitochondrial ND1 (subunit one of NADH dehydrogenase). Neither a close proximity species based on the gene encoding the mitochondrial COI (cytochrome oxidase subunit one) could be found in publicly accessible nucleotide databases. While the cryptic H. stubbei sp. nov. reveals no obvious cranial and morphological differences, few external characteristics are dissimilar to both H. alaschanicus and H. savii (caucasicus). Currently, Hypsugo stubbei sp. nov. was found at four different locations in Mongolia. Among the 11 specimens captured, six facilitated a genetic assignment. Based on the current scarce data records, the species seems to occur mainly in the far west of Mongolia inhabiting semi-deserts and steppes up to high mountain areas. An overlapping distribution with H. alaschanicus cannot be excluded based on the limited data currently available

A multi-gene phylogeny of Lactifluus (Basidiomycota, Russulales) translated into a new infrageneric classification of the genus

Infrageneric relations of the genetically diverse milkcap genus Lactifluus (Russulales, Basidiomycota) are poorly known. Currently used classification systems still largely reflect the traditional, mainly morphological, characters used for infrageneric delimitations of milkcaps. Increased sampling, combined with small-scale molecular studies, show that this genus is underexplored and in need of revision. For this study, we assembled an extensive dataset of the genus Lactifluus, comprising 80 % of all known species and 30 % of the type collections. To unravel the infrageneric relationships within this genus, we combined a multi-gene molecular phylogeny, based on nuclear ITS, LSU, RPB2 and RPB1, with a morphological study, focussing on five important characteristics (fruit body type, presence of a secondary velum, colour reaction of the latex/context, pileipellis type and presence of true cystidia). Lactifluus comprises four supported subgenera, each containing several supported clades. With extensive sampling, ten new clades and at least 17 new species were discovered, which highlight the high diversity in this genus. The traditional infrageneric classification is only partly maintained and nomenclatural changes are proposed. Our morphological study shows that the five featured characteristics are important at different evolutionary levels, but further characteristics need to be studied to find morphological support for each clade. This study paves the way for a more detailed investigation of biogeographical history and character evolution within Lactifluus

Selten in der Mongolei nachgewiesene Fledermausarten

In der Mongolei sind gegenwärtig 20 verschiedene Fledermausarten aus sieben Gattungen nachgewiesen worden. Hierbei handelt es sich um sieben Arten Myotis, vier Arten Plecotus, drei Arten Eptesicus, zwei Arten Vespertilio, zwei Arten Hypsugo und je eine Art der Gattungen Nyctalus und Murina. Von den bisher bekannten Species konnten 11 nur selten und zwei von ihnen nur einmal gefunden worden. Bis auf die Arten der Gattung Plecotus und Hypsugo, die in getrennten Publikationen in diesem Heft behandelt werden (DOLCH et al. 2021a, 2021b), sind nachfolgend diese seltenen Funde zusammengestellt. Die Grundlage für die Häufigkeitseinschätzung bilden über 2000 Fledermausfänge an über 100 Fangplätzen der Feldexpeditionen „Chiroptera Mongolia“ (seit 1999) und der Mongolisch-Deutschen Biologischen Expeditionen, die seit 1962 unter der Leitung von Michael Stubbe in unterschiedlicher Zusammensetzung durchgeführt wurden

The taxonomy of the Trichophyton rubrum complex: a phylogenomic approach

The medically relevant Trichophyton rubrum species complex has a variety of phenotypic presentations but shows relatively little genetic differences. Conventional barcodes, such as the internal transcribed spacer (ITS) region or the beta-tubulin gene, are not able to completely resolve the relationships between these closely related taxa. T. rubrum, T. soudanense and T. violaceum are currently accepted as separate species. However, the status of certain variants, including the T. rubrum morphotypes megninii and kuryangei and the T. violaceum morphotype yaoundei, remains to be deciphered. We conducted the first phylogenomic analysis of the T. rubrum species complex by studying 3105 core genes of 18 new strains from the BCCM/IHEM culture collection and nine publicly available genomes. Our analyses revealed a highly resolved phylogenomic tree with six separate clades. Trichophyton rubrum, T. violaceum and T. soudanense were confirmed in their status of species. The morphotypes T. megninii, T. kuryangei and T. yaoundei all grouped in their own respective clade with high support, suggesting that these morphotypes should be reinstituted to the species-level. Robinson-Foulds distance analyses showed that a combination of two markers (a ubiquitin-protein transferase and a MYB DNA-binding domain-containing protein) can mirror the phylogeny obtained using genomic data, and thus represent potential new markers to accurately distinguish the species belonging to the T. rubrum complex