Open Access

Alexei Kudryashov

• In this thesis we demonstrate a universal property of symmetric L-theory as a space-valued functor from the category of Waldhausen categories with Spanier Whitehead products in the sense of Weiss-Williams. Specifically, we characterise symmetric L-Theory as the target of the ``universal bordism characteristic of symmetric Poincaré objects". Furthermore, we show that the construction of Quinn's bordism spaces of ad theories in the sense of Laures-McClure satisfies an analogous characterisation. The main novel ingredient of our work is the development of a simple abstract setting for universality that unifies both examples. There are two parts to this thesis: Part I establishes the abstract foundations and describes applications. Part II is a technical extension of the first part, based on a further analysis of sufficient conditions for universal bordism characteristics and the problem of how to extend their targets to spectrum-valued functors in a natural way. We introduce a secondIn this thesis we demonstrate a universal property of symmetric L-theory as a space-valued functor from the category of Waldhausen categories with Spanier Whitehead products in the sense of Weiss-Williams. Specifically, we characterise symmetric L-Theory as the target of the ``universal bordism characteristic of symmetric Poincaré objects". Furthermore, we show that the construction of Quinn's bordism spaces of ad theories in the sense of Laures-McClure satisfies an analogous characterisation. The main novel ingredient of our work is the development of a simple abstract setting for universality that unifies both examples. There are two parts to this thesis: Part I establishes the abstract foundations and describes applications. Part II is a technical extension of the first part, based on a further analysis of sufficient conditions for universal bordism characteristics and the problem of how to extend their targets to spectrum-valued functors in a natural way. We introduce a second more specialised framework for this investigation and illustrate the theory in two explicit examples; namely, Quinn's Bordism machine of ad theories and symmetric L-theory in the setting of additive categories with chain duality introduced by A. A. Ranicki.…