The weak choice principle WISC may fail in the category of sets

The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos as defined by van den Berg.Comment: v2 Change of title and abstract; v3 Almost completely rewritten after referee pointed out critical mistake. v4 Final version. Will be published in Studia Logica. License is CC-B

On the existence of bibundles

We consider the existence of bibundles, in other words locally trivial principal $G$ spaces with commuting left and right $G$ actions. We show that their existence is closely related to the structure of the group \Out(G) of outer automorphisms of $G$. We also develop a classifying theory for bibundles. The theory is developed in full generality for $(H, G)$ bibundles for a crossed-module $(H, G)$ and we show with examples the close links with loop group bundles.Comment: This version accepted to appear in the Proceedings of the London Mathematical Society. Various changes made at referee's suggestio

Extending Whitney's extension theorem: nonlinear function spaces

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where $C$ only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition A.3, which makes no impact on proofs or results. Version submitted for publication. v3 small changes in response to referee comments, title extended. v4 crucial gap filled, results not affected. v5 final version to appear in Annales de l'Institut Fourie

Design and Implementation of Rotational Print Heads to Control Fiber Orientation in Material Extrusion Additive Manufacturing

Short fibers are being incorporated into 3D printable polymers to modify the properties of the printed composite. These fibers are, however, highly aligned in the print direction limiting the range of programmable site-specific characteristics. Recent work has demonstrated a novel approach of adding an additional shear field, through rotation, during the direct ink writing process, enabling control over the fiber orientation. This work endeavors to reproduce those results and demonstrate the same novel concept in fused filament fabrication, and outlines the challenges using these two systems

A topological fibrewise fundamental groupoid

It is well-known that for certain local connectivity assumptions the fundamental groupoid of a topological space can be equipped with a topology making it a topological groupoid. In other words, the fundamental groupoid functor can be lifted through the forgetful functor from topological groupoids to groupoids. This article shows that for a map $Y \to X$ with certain relative local connectivity assumptions, the fibrewise fundamental groupoid can also be lifted to a topological groupoid over the space $X$. This allows the construction of a simply-connected covering space in the setting of fibrewise topology, assuming a local analogue of the definition of an ex-space. When applied to maps which are up-to-homotopy locally trivial fibrations the result is a categorified version of a covering space. The fibrewise fundamental groupoid can also be used to define a topological fundamental bigroupoid of a (suitably locally connected) topological space.Comment: Typos fixed and references/corrections added on referee's and editor's comments. Released under a CC0 license (http://creativecommons.org/publicdomain/zero/1.0/

The elementary construction of formal anafunctors

This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully faithful.Comment: v1: 20 pages, many diagrams. v2: updated after referee comments, 32 pages. v3: one minor terminology change, one proof streamlined after additional feedback, 32 page