The cotangent complex and Thom spectra

We first prove, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then prove the following theorem: if $R$ is an $E_\infty$-ring spectrum and $f:G\to \mathrm{Pic}(R)$ is a map of $E_\infty$-groups, then the cotangent complex over $R$ of the Thom $E_\infty$-$R$-algebra of $f$ is equivalent to the smash product of $Mf$ and the connective spectrum associated to $G$.Comment: 19 page

Shadows are Bicategorical Traces

The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to bicategories, we prove that there is an equivalence between functors out of Hochschild homology of a bicategory and shadows on that bicategory. As part of that proof we give a computational description of pseudo-functors out of the truncated simplex category and a variety thereof, which can be of independent interest. Building on this result we present three applications: (1) We provide a new, conceptual proof that shadows, and as a result the Euler characteristic and trace introduced by Campbell and Ponto, are Morita invariant. (2) We strengthen this result by using an explicit computation of the Hochschild homology of the free adjunction bicategory to show that the construction of the Euler characteristic is homotopically unique. (3) We generalize the construction of $\mathscr{V}$-enriched Hochschild homology, where $\mathscr{V}$ is a presentably symmetric monoidal $\infty$-category, to bimodules, and prove it gives us a shadow.Comment: 55 Pages, comments welcome