On endomorphisms of topological Hochschild homology

TL;DR

This work determines the endomorphism spectra of THH viewed as a functor on stable \(\infty\)-categories, as well as THH relative to a base ring spectrum and THH with $\mathcal{O}$-multiplicative structures. It shows that plain endomorphisms are governed by the circle action, with $\mathrm{end}(\mathrm{THH}) \simeq \mathbb{S}[S^1]$ and multiplicative endomorphisms given by $\operatorname{Map}_{\mathrm{CAlg}(\mathrm{Sp})}(\mathbb{S}^{S^1}, \mathbb{S})$, whose $\pi_0$ is the profinite integers and which admit rational/localized refinements. The approach hinges on Day convolution for localizing invariants and a generalized Dundas–McCarthy description of THH in terms of $K$-theory, enabling explicit computation of endomorphisms and revealing when extra operations appear under localization. In the operadic setting, endomorphisms of THH as a functor on $\mathrm{Alg}_{\mathcal{O}}(\mathrm{Cat}^{\mathrm{perf}})$ are controlled by a direct sum over arities, and localization (rational or chromatic) yields a clean description, while integral cases exhibit additional cyclotomic phenomena and truly new operations.

Abstract

We compute endomorphisms of topological Hochschild homology ($\mathrm{THH}$) as a functor on stable $\infty$-categories, as well as variants thereof: we also compute endomorphisms of the $k$-linear Hochschild homology functor $\mathrm{HH}_k$ over some base $k$; and endomorphisms of $\mathrm{THH}$ as a functor on stably symmetric monoidal $\infty$-categories.

Theorems & Definitions (100)

• Theorem 1
• Theorem 2
• Proposition 1.1
• Remark 1.2
• Remark 1.3
• Lemma 1.4
• proof
• Corollary 1.6
• Corollary 1.8
• Remark 1.9