Comorphisms, Duality, and L-infinity Comorphisms

Abstract

This thesis details homotopy bracket analogues of duality between Lie algebroids and Poisson manifolds established by Higgins-Mackenzie. This is achieved through Voronov's thick morphisms, together with several new constructions; most notably, we introduce L-infinity comorphisms of L-infinity 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 L-infinity algebras and L-infinity algebroids, and investigate the relationship between an L-infinity algebroid E and its manifestations on the neighbouring bundles E, E*, \Pi E, and \Pi E*. In Chapter 4, we introduce the category of L-infinity comorphisms of L-infinity algebroids. The central achievements of the thesis, Theorem 15 and Theorem 16, establish that this category is equivalent to the categories of factorisable P-infinity and S-infinity thick morphisms, which are homotopy analogues of Poisson maps of Poisson bundles. This result generalises the previously mentioned duality relationships of Higgins-Mackenzie. 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)