Day convolution and the Hodge filtration on THH

Abstract

This thesis is divided into two chapters. In the first, given symmetric monoidal oc-categories C and D, subject to mild hypotheses on D, we define an oc-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). In the second, we develop a Hodge filtration on the topological Hochschild homolgy spectrum of a commutative ring spectrum and describe its elementary properties.