The \'etale symmetric K\"unneth theorem

Let $k$ be an algebraically closed field, $l\neq\operatorname{char} k$ a prime number, and $X$ a quasi-projective scheme over $k$. We show that the \'etale homotopy type of the $d$th symmetric power of $X$ is $\mathbb Z/l$-homologically equivalent to the $d$th strict symmetric power of the \'etale homotopy type of $X$. We deduce that the $\mathbb Z/l$-local \'etale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.Comment: revised version, comments welcome

Formality of P\mathbb{P}-objects

We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.Comment: 23 pages, many changes to improve presentation, strengthened results in Section 5, same content as published versio

The Galois group of a stable homotopy theory

To a "stable homotopy theory" (a presentable, symmetric monoidal stable $\infty$-category), we naturally associate a category of finite \'etale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. We then calculate the Galois groups in several examples. For instance, we show that the Galois group of the periodic $\mathbf{E}_\infty$-algebra of topological modular forms is trivial and that the Galois group of $K(n)$-local stable homotopy theory is an extended version of the Morava stabilizer group. We also describe the Galois group of the stable module category of a finite group. A fundamental idea throughout is the purely categorical notion of a "descendable" algebra object and an associated analog of faithfully flat descent in this context.Comment: 93 pages. To appear in Advances in Mathematic

Constructible sheaves on schemes and a categorical K\"unneth formula

We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in condensed rings. We use it to prove a K\"unneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic $p>0$ for a wide variety of coefficients, including finite discrete rings, algebraic field extensions $E\supset \mathbb{Q}_\ell$, $\ell\neq p$ and their rings of integers $\mathcal{O}_E$. When combined with results of Xue, this applies to the cohomology of moduli stacks of shtukas over global function fields.Comment: Comments welcome