AI 中文总结
本文提出了一种构造VB 2-群体的2-余切的方法，并展示了其在李2-群体上的应用，包括对2-移位辛结构的表征和计算。
AI 中文摘要
本文定义了李\(n -\)群胚上\(VB\)\(n -\)群胚的\(n -\)对偶，并研究其性质。\(n = 0\)时是对偶向量丛构造，\(n = 1\)时是普拉迪内斯关于李群胚上\(VB\)群胚对偶的构造，包括科斯特、达佐德和温斯坦的余切辛群胚。\(n = 2\)时，提出新构造表明\(VB\)\(2 -\)群胚存在\(VB\)\(2 -\)对偶且自身也是\(VB\)\(2 -\)群胚。其典范对偶配对在同伦意义下非退化。还将此应用于李\(2 -\)群胚的切丛得到余切\(VB\)\(2 -\)群胚，以两种方式应用此构造：一是将李\(2 -\)群胚上的\(2 -\)移位辛结构刻画为其切丛与\(2 -\)余切群胚之间的莫里塔等价；二是计算李\(1 -\)群胚的\(2 -\)余切并表明它与\(1 -\)余切的棒构造是辛莫里塔等价。还发展了单纯向量空间的\(n -\)对偶理论，通过重新表述艾伦伯格 - 齐尔伯定理得到典范\(n -\)对偶配对在同伦意义下对所有\(n -\)型是非退化的。
英文摘要
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.
CommentsPhD Thesis at the University of Göttingen, defended May 19th 2025. 200 pages, 9 figures
