The Loday Construction on Hopf Algebras

Topological Hochschild homology can be viewed as a special case of the more general Loday construction. This is known to not be a stable invariant using a counterexample by Dundas-Tenti employing the stably splitting of a torus into a wedge of spheres. However, while stability for the Loday construction does not hold in general, extra structure on the input ring spectrum can guarantee stability nonetheless. For example, Berest-Ramadoss-Yeung proved that stability holds for Hopf algebras by relating the Loday construction to representation homology. In this talk I will explain a direct categorical proof of this fact, which avoids representation homology, using the framework of infinity categories. The result is part of a “Women in Topology” project on the stability of the Loday construction together with S. Klanderman, A. Lindenstrauss, B. Richter, and F. Zou.