Categorified Algebra and Equivariant Homotopy Theory

This dissertation comprises three collections of results, all united by a common theme. The theme is the study of categories via algebraic techniques, considering categories themselves as algebraic objects. This algebraic approach to category theory is central to noncommutative algebraic geometry, as realized by recent advances in the study of noncommutative motives.

We have success proving algebraic results in the general setting of symmetric monoidal and semiring infinity categories, which categorify abelian groups and rings, respectively. For example, we prove that modules over the semiring category Fin of finite sets are cocartesian monoidal infinity categories, and modules over Burn (the Burnside infinity category) are additive infinity categories.

As a consequence, we can regard Lawvere theories as cyclic Fin^op-modules, leading to algebraic foundations for the higher categorical study of Lawvere theories. We prove that Lawvere theories function as a home for an algebraic Yoneda lemma.

Finally, we provide evidence for a formal duality between naive and genuine equivariant homotopy theory, in the form of a group-theoretic Eilenberg-Watts Theorem. This sets up a parallel between equivariant homotopy theory and motivic homotopy theory, where Burnside constructions are analogous to Morita theory. We conjecture that this relationship could be made precise within the context of noncommutative motives over the field with one element.

In fact, the connections equivariant homotopy theory and the field with one element recur throughout the thesis. There are promising suggestions that each of these two subjects can be advanced by further work in this area of algebraic category theory.