Duals of higher vector bundles and cotangents of Lie 2-groupoids

In this thesis we define $n$-duals of VB $n$-groupoids over Lie $n$-groupoids and study their properties. For $n = 0$ this returns the dual vector bundle construction, while for $n = 1$ this returns Pradines's construction of the dual of a VB groupoid over a Lie groupoid, which includes the cotangent symplectic groupoid of Coste, Dazord and Weinstein. For $n = 2$, we propose a new construction that shows that VB 2-duals exist for VB 2-groupoids and they are VB 2-groupoids themselves. Their canonical dual pairings are nondegenerate up to homotopy in the same sense as shifted symplectic structures. In particular, we can apply this construction to the tangent of a Lie 2-groupoid and obtain a cotangent VB 2-groupoid (the 2-cotangent) which is canonically 2-shifted symplectic. We apply this in two ways: First, to characterize 2-shifted symplectic structures on a Lie 2-groupoid as Morita equivalences between its tangent and 2-cotangent groupoid. Second, to compute the 2-cotangent of a Lie 1-groupoid and show it is symplectic Morita equivalent to the bar construction of the 1-cotangent. Along the way, we develop the theory of $n$-duals for simplicial vector spaces, which covers the case where the base is a point. In this case, $n$-duals always exist, as they are defined by a mapping space construction. By a reformulation of the Eilenberg-Zilber theorem in terms of mapping spaces, we obtain that the canonical $n$-dual pairing is nondegenerate up to homotopy for all $n$-types.