On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces

We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G, the weak homotopy type of X agrees with that of BG. Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction of an almost free action of a Lie group G on a smooth manifold M as the classifying space of a category whose objects consists of smooth maps R^n to M which are transverse to all the G-orbits, where n=dim M - dim G. We also prove a generalization of Segal's theorem, which presents the weak homotopy type of Haefliger's groupoid $\Gamma^q$ as the classifying space of the monoid of self-embeddings of R^q, and our generalization gives analogous presentations for the weak homotopy type of the Lie groupoids $\Gamma^{Sp}_{2q}$ and $R\Gamma^q$ which are related to the classification of foliations with transverse symplectic forms and transverse metrics respectively. We also give a short and simple proof of Segal's original theorem using our machinery.Comment: 47 page

Compactly Generated Stacks: A Cartesian Closed Theory of Topological Stacks

A convenient bicategory of topological stacks is constructed which is both complete and Cartesian closed. This bicategory, called the bicategory of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of bicategories between compactly generated stacks and those classical topological stacks which admit locally compact Hausdorff atlases. Compactly generated stacks are also equivalent to a bicategory of topological groupoids and principal bundles, just as in the classical case. If a classical topological stack and a compactly generated stack have a presentation by the same topological groupoid, then they restrict to the same stack over locally compact Hausdorff spaces and are homotopy equivalent.Comment: 51 pages. Fixed some technical point-set topology errors from the previous version as well as errors involving the homotopy type of compactly generated stack

Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi

We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring spectra as in Lurie's work, but also to differential topology, complex geometry, the theory of supermanifolds, derived manifolds etc., where it produces a theory of higher generalized orbifolds appropriate for these settings. This universal framework yields new insights into the general theory of Deligne-Mumford stacks and orbifolds, including a representability criterion which gives a categorical characterization of such generalized Deligne-Mumford stacks. This specializes to a new categorical description of classical Deligne-Mumford stacks, a result sketched in previous work of the author, which extends to derived and spectral Deligne-Mumford stacks as well.Comment: 121 pages. Fixed some minor errors in the example

Fatal Exudative Dermatitis (FED) in Island Populations of Red Squirrels (Sciurus vulgaris): Spillover of a Virulent Staphylococcus aureus Clone (ST49) From Reservoir Hosts.

Fatal exudative dermatitis (FED) is a significant cause of death of red squirrels (Sciurus vulgaris) on the island of Jersey in the Channel Islands where it is associated with a virulent clone of Staphylococcus aureus, ST49. S. aureus ST49 has been found in other hosts such as small mammals, pigs and humans, but the dynamics of carriage and disease of this clone, or any other lineage in red squirrels, is currently unknown. We used whole-genome sequencing to characterize 228 isolates from healthy red squirrels on Jersey, the Isle of Arran (Scotland) and Brownsea Island (England), from red squirrels showing signs of FED on Jersey and the Isle of Wight (England) and a small number of isolates from other hosts. S. aureus was frequently carried by red squirrels on the Isle of Arran with strains typically associated with small ruminants predominating. For the Brownsea carriage, S. aureus was less frequent and involved strains associated with birds, small ruminants and humans, while for the Jersey carriage S. aureus was rare but ST49 predominated in diseased squirrels. By combining our data with publicly available sequences, we show that the S. aureus carriage in red squirrels largely reflects frequent but facile acquisitions of strains carried by other hosts sharing their habitat ('spillover'), possibly including, in the case of ST188, humans. Genome-wide association analysis of the ruminant lineage ST133 revealed variants in a small number of mostly bacterial-cell-membrane-associated genes that were statistically associated with squirrel isolates from the Isle of Arran, raising the possibility of specific adaptation to red squirrels in this lineage. In contrast there is little evidence that ST49 is a common carriage isolate of red squirrels and infection from reservoir hosts such as bank voles or rats, is likely to be driving the emergence of FED in red squirrels

Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes

For a log scheme locally of finite type over C, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over C, another natural candidate is the profinite \'etale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over C, these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite \'etale homotopy type of its infinite root stack