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We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's ''The volume of a differentiable stack''. [ProQuest: [...] denotes formulae omitted.]

In previous work by the first two authors, Frobenius and commutative algebra objects in the category of spans of sets were characterized in terms of simplicial sets satisfying certain properties. In this paper, we find a similar characterization for the analogous coherent structures in the bicategory of spans of sets. We show that commutative and Frobenius pseudomonoids in \\(\\operatorname{Span}\\) correspond, respectively, to paracyclic sets and \\(\\Gamma\\)-sets satisfying the \\(2\\)-Segal conditions. These results connect closely with work of the third author on \\(A_\\infty\\) algebras in \\(\\infty\\)-categories of spans, as well as the growing body of work on higher Segal objects. Because our motivation comes from symplectic geometry and topological field theory, we emphasize the direct and computational nature of the classifications and their proofs.

We present general reduction procedures for Courant, Dirac and generalized complex structures, in particular when a group of symmetries is acting. We do so by taking the graded symplectic viewpoint on Courant algebroids and carrying out graded symplectic reduction, both in the coisotropic and hamiltonian settings. Specializing the latter to the exact case, we recover in a systematic way the reduction schemes of Bursztyn-Cavalcanti-Gualtieri.

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids. Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

We give some new examples of Frobenius objects in the category of sets and relations \\(\\textbf{Rel}\\). One example is a groupoid with a twisted counit. Another example is the set of conjugacy classes of a group. We also classify Frobenius objects in \\(\\textbf{Rel}\\) with two or three elements, and we compute the associated surface invariants using the partition functions of the corresponding TQFTs.

We give an explicit description, in terms of bracket, anchor, and pairing, of the standard cochain complex associated to a Courant algebroid. In this formulation, the differential satisfies a formula that is formally identical to the Cartan formula for the de Rham differential. This perspective allows us to develop the theory of Courant algebroid connections in a way that mirrors the classical theory of connections. Using a special class of connections, we construct secondary characteristic classes associated to any Courant algebroid.

We give a characterization, in terms of simplicial sets, of Frobenius objects in the category of relations. This result generalizes a result of Heunen, Contreras, and Cattaneo showing that special dagger Frobenius objects in the category of relations are in correspondence with groupoids. As an additional example, we construct a Frobenius object in the category of relations whose elements are certain cohomology classes in a compact oriented Riemannian manifold.

Hypokalemic periodic paralysis (HPP) is an uncommon condition resulting from channelopathy, impacting skeletal muscles. It is distinguished by episodes of sudden and temporary muscle weakness alongside low potassium levels. The normalization of potassium resolves the associated paralysis. Most of these cases are hereditary. Few cases are acquired and are associated with an etiology related to endocrine disorders (e.g., thyrotoxicosis, hyperaldosteronism, and hypercortisolism). It is characterized by acute flaccid paralysis, usually of the ascending type, affecting the proximal region more than the distal region. Herein, we report the case of a 29-year-old male who instead of the ascending type presented with descending-type acute flaccid paralysis. Potassium level at presentation was 1.7 mEq/L. The patient was managed with parenteral and oral potassium supplementation, after which the weakness was completely resolved.

We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.

We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's \"The volume of a differentiable stack\".