This thesis details homotopy bracket analogues of duality between Lie algebroids and Poisson manifolds established by HigginsMackenzie. This is achieved through Voronov's thick morphisms, together with several new constructions; most notably, we introduce Linfinity comorphisms of Linfinity algebroids. We review the theory of Lie algebroids and Poisson manifolds in Chapter 2, introducing Vaintrob's notion of a homological vector field, and demonstrate its use by giving alternative proofs of some classical results. In Chapter 3, we turn our attention to Linfinity algebras and Linfinity algebroids, and investigate the relationship between an Linfinity algebroid E and its manifestations on the neighbouring bundles E, E*, \Pi E, and \Pi E*. In Chapter 4, we introduce the category of Linfinity comorphisms of Linfinity algebroids. The central achievements of the thesis, Theorem 15 and Theorem 16, establish that this category is equivalent to the categories of factorisable Pinfinity and Sinfinity thick morphisms, which are homotopy analogues of Poisson maps of Poisson bundles. This result generalises the previously mentioned duality relationships of HigginsMackenzie. In Chapter 5, we introduce categories of fibrewise linear thick morphisms. These are more general than the factorisable thick morphisms of the previous chapter. We argue that these categories are the appropriate generalisation of the category of vector bundles to the thick setting, as by Proposition 10, these categories admit tangent and antitangent functors from the categories of thick morphisms. We complete the theory with the introduction of the categories of thick comorphisms, which are generalisations of vector bundle comorphisms, and show that these categories are equivalent to the categories of fibrewise linear thick morphisms. Using these notions, we end with Theorem 18, which is a thick analogue of Higgins and Mackenzie's Theorem 7, regarding the cotangent comorphism of a Poisson map.
Date of Award  1 Aug 2024 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Theodore Voronov (Supervisor) 

 differential geometry
 supergeometry
 poisson geometry
 higher bracket
 comorphism
 homotopy poisson
 linfinity algebroid
 bracket geometry
Comorphisms, Duality, and Linfinity Comorphisms
Brady, S. (Author). 1 Aug 2024
Student thesis: Phd