We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of “stability relative to a class of functors”, which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.

Let $\text{K}\left({\mathbb{𝔽}}_q\right)$ be the algebraic $K$-theory spectrum of the finite field with $q$ elements and let $p\ge 5$ be a prime number coprime to $q$. We study the mod $p$ and $v_1$ topological Hochschild homology of $\text{K}\left({\mathbb{𝔽}}_q\right)$, denoted $V{\left(1\right)}_{\ast }\text{THH}\left(\text{K}\left({\mathbb{𝔽}}_q\right)\right)$, as an ${\mathbb{𝔽}}_p$-algebra. The computations are organized in four different cases, depending on the $p$-adic behavior of the function $q^n-1$. We use several different spectral sequences, in particular the Bökstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We calculate the ${\mathbb{𝔽}}_p$-algebra ${\text{THH}}_{\ast }\left(\text{K}\left({\mathbb{𝔽}}_q\right);H{\mathbb{𝔽}}_p\right)$, and we compute $V{\left(1\right)}_{\ast }\text{THH}\left(\text{K}\left({\mathbb{𝔽}}_q\right)\right)$ in the first two cases.

In previous work, we developed a generalized Waldhausen $S_{•}$-construction whose input is an augmented stable double Segal space and whose output is a $2$-Segal space. Here, we prove that this construction recovers the previously known $S_{•}$-constructions for exact categories and for stable and exact $\left(\infty ,1\right)$-categories, as well as the relative $S_{•}$-construction for exact functors.

We use descent theoretic methods to solve the homotopy limit problem for Hermitian $K$-theory over quasicompact and quasiseparated base schemes. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermitian $K$-theory over $\text{Spec}\left(ℂ\right)$, showing that the only invertible cellular spectra are suspensions of the tensor unit.