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A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are ''good''. We then apply this notion to the category [...] of linear canonical relations and the result [...] of our version of the WW construction, identifying the morphisms in the latter with pairs [...] consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in [...] itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts. [ProQuest: [...] denotes formulae omitted.]

Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G -bundle, P , over a compact oriented surface with boundary, , carries a Poisson structure. If we trivialize P over a finite number of points on then the moduli space carries a quasi-Poisson structure instead. Our first result is to describe this quasi-Poisson structure in terms of an intersection form on the fundamental groupoid of the surface, generalizing results of Massuyeau and Turaev citeMassuyeau:2012uw,Turaev:2007jh. Our second result is to extend this framework to quilted surfaces, i.e. surfaces where the structure group varies from region to region and a reduction (or relation) of structure occurs along the borders of the regions, extending results of the second author citeSevera:2011ug,Severa98,Severa:2005vla. We describe the Poisson structure on the moduli space for a quilted surface in terms of an operation on spin networks, i.e. graphs immersed in the surface which are endowed with some additional data on their edges and vertices. This extends the results of various authors citeGoldman:1986eh,Goldman:1984hr,Roche:2000ws,Andersen:1996ur.

It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid \\(A\\): The source-simply connected Lie groupoid \\(G M\\) integrating \\(A\\) automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the `quasi' settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs \\((E,A)\\), thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid \\(G\\) integrating \\(A\\) is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism.

It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid \\(A\\): The source-simply connected Lie groupoid \\(G M\\) integrating \\(A\\) automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the `quasi' settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs \\((E,A)\\), thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid \\(G\\) integrating \\(A\\) is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism.

In this thesis we develop the notion of [special characters omitted]-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q-Poisson ([special characters omitted])-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-Lagrangian) subbundles W ⊆ [special characters omitted] of a Courant algebroid such that the Courant bracket endows W naturally with the structure of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the theory of q-Poisson ([special characters omitted])-structures.

A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are \"good\". We then apply this notion to the category \\(SLREL\\) of linear canonical relations and the result \\( WW(SLREL)\\) of our version of the WW construction, identifying the morphisms in the latter with pairs \\((L,k)\\) consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in \\(SLREL\\) itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.

In this paper, we show that two constructions form stacks: Firstly, as one varies the \\(ınfty\\)-topos, \\(X\\), Lurie's homotopy theory of higher categories internal to \\(X\\) varies in such a way as to form a stack over the \\(ınfty\\)-category of all \\(ınfty\\)-topoi. Secondly, we show that Haugseng's construction of the higher category of iterated spans in a given \\(ınfty\\)-topos (equipped with local systems) can be used to define various stacks over that \\(ınfty\\)-topos. As a prerequisite to these results, we discuss properties which limits of \\(ınfty\\)-categories inherit from the \\(ınfty\\)-categories comprising the diagram. For example, Riehl and Verity have shown that possessing (co)limits of a given shape is hereditary. Extending their result somewhat, we show that possessing Kan extensions of a given type is heriditary, and more generally that the adjointability of a functor is heriditary.

We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to connect our work with more traditional approaches, and also to put it into a wider context suggesting possible generalizations.

A Dirac structure is a Lagrangian subbundle of a Courant algebroid, \\(L\\), which is involutive with respect to the Courant bracket. In particular, \\(L\\) inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, \\(W\\), together with a pseudo-connection on its sections, satisfying a natural integrability condition. As a consequence of the definition, \\(W\\) will be a Lie algebroid. Allowing non-isotropic subbundles of \\(E\\) incorporates non-skew tensors and connections into Dirac geometry. Novel examples of pseudo-Dirac structures arise in the context of quasi-Poisson geometry, Lie theory, generalized Kähler geometry, and Dirac Lie groups, among others. Despite their greater generality, we show that pseudo-Dirac structures share many of the key features of Dirac structures. In particular, they behave well under composition with Courant relations.

In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [26] inverts our integration.