arXiv subjects
Publications and source records attributed to Saul Glasman.
We show that the algebraic K-theory space of stable infinity-categories is canonically functorial in polynomial functors. As a consequence, we obtain a new proof of Bökstedt's calculation of $\mathrm{THH}(\mathbb{F}_p)$.
Let $X$ be a quasicompact quasiseparated scheme. Write $\operatorname{Gal}(X)$ for the category whose objects are geometric points of $X$ and whose morphisms are specializations in the étale topology. We define a natural profinite topology on the category $\operatorname{Gal}(X)$ that globalizes the topologies of the absolute Galois groups of the residue fields of the points of $X$. One of the main results of this book is that $\operatorname{Gal}(X)$ variant of MacPherson's exit-path category suitable for the étale topology: we construct an equivalence between representations of $\operatorname{Gal}(X)$ and constructible sheaves on $X$. We show that this 'exodromy equivalence' holds with nonabelian coefficients and with finite abelian coefficients. More generally, by using the pyknotic/condensed formalism, we extend this equivalence to coefficients in the category of modules over profinite rings and algebraic extensions of $\mathbf{Q}_{\ell}$. As an 'exit-path category', the topological category $\operatorname{Gal}(X)$ also gives rise to a new, concrete description of the étale homotopy type of $X$. We also prove a higher categorical form of Hochster Duality, which reconstructs the entire étale topos of a quasicompact and quasiseparated scheme from the topological category $\operatorname{Gal}(X)$. Appealing to Voevodsky's proof of a conjecture of Grothendieck, we prove the following reconstruction theorem for normal varieties over a finitely generated field $k$ of characteristic $0$: the functor $X\mapsto\operatorname{Gal}(X)$ from normal $ k $-varieties to topological categories with an action of $\operatorname{G}_{k}$ and equivariant functors that preserve minimal objects is fully faithful.
For any $E_\infty$ ring spectrum $E$, we show that there is an algebra $\mathrm{Pow}(E)$ of stable power operations that acts naturally on the underlying spectrum of any $E$-algebra. Further, we show that there are maps of rings $E \to \mathrm{Pow}(E) \to \mathrm{End}(E)$, where the latter determines a restriction from power operations to stable operations in the cohomology of spaces. In the case where $E$ is the mod-$p$ Eilenberg-Mac Lane spectrum, this realizes a natural quotient from Mandell's algebra of generalized Steenrod operations to the mod-$p$ Steenrod algebra. More generally, this arises as part of a classification of endomorphisms of representable functors from an $\infty$-category $\mathcal{C}$ to spectra, with particular attention to the case where $\mathcal{C}$ is an $\mathcal{O}$-monoidal $\infty$-category.
We show that the category of $n$-excisive functors from the $\infty$-category of spectra to a target stable $\infty$-category $\mathbf{E}$ is equivalent to the category of $\mathbf{E}$-valued Mackey functors on an indexing category built from finite sets and surjections. This new classification of polynomial functors arises from an investigation of the structure present on cross effects. The path to this result involves a pair of surprising extension theorems for polynomial functors and a discussion of some interesting topics in semiadditive $\infty$- category theory, including a formula for the free semiadditive $\infty$-category on an $\infty$-category.
We develop a theory of Mackey functors on epiorbital categories which simultaneously generalizes the theory of genuine $G$-spectra for a finite group $G$ and the theory of $n$-excisive functors on the category of spectra. Using a new theory of stratifications of a stable $\infty$-category along a finite poset, we prove a simultaneous generalization of two reconstruction theorems: one by Abram and Kriz on recovering $G$-spectra from structure on their geometric fixed point spectra for abelian $G$, and one by Arone and Ching that recovers an $n$-excisive functor from structure on its derivatives. We deduce a strong tom Dieck splitting theorem for $K(n)$-local $G$-spectra and reprove a theorem of Kuhn on the $K(n)$-local splitting of Taylor towers.
We solve a problem proposed by Khovanov by constructing, for any set of primes $S$, a triangulated category (in fact a stable $\infty$-category) whose Grothendieck group is $S^{-1}\mathbf{Z}$. More generally, for any exact $\infty$-category $E$, we construct an exact $\infty$-category $S^{-1}E$ of equivariant sheaves on the Cantor space with respect to an action of a dense subgroup of the circle. We show that this $\infty$-category is precisely the result of categorifying division by the primes in $S$. In particular, $K_n(S^{-1}E)\cong S^{-1}K_n(E)$.
We introduce the study of parametrized higher category theory and parametrized higher algebra, and we describe the main theorems of the series of Exposés that make up the monograph.
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor $\infty$-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straightening. The key results of this Exposé are: (1) a universal characterization of the $T$-$\infty$-category of $T$-objects in any $\infty$-category, (2) the existence of an internal Hom for $T$-$\infty$-categories, and (3) a parametrized Yoneda lemma.
Effective Burnside $\infty$-categories are the centerpiece of the $\infty$-categorical approach to equivariant stable homotopy theory. In this étude, we recall the construction of the twisted arrow $\infty$-category, and we give a new proof that it is an $\infty$-category, using an extremely helpful modification of an argument due to Joyal--Tierney. The twisted arrow $\infty$-category is in turn used to construct the effective Burnside $\infty$-category. We employ a variation on this theme to construct a fibrewise effective Burnside $\infty$-category. To show that this constuctionworks fibrewise, we introduce a fragment of a theory of what we call marbled simplicial sets, and we use a yet further modified form of the Joyal--Tierney argument.
In this short étude, we observe that the full structure of a recollement on a stable infinity-category can be reconstructed from minimal data: that of a reflective and coreflective full subcategory. The situation has more symmetry than one would expect at a glance. We end with a practical lemma on gluing equivalences along a recollement.
With an explicit, algebraic indexing $(2,1)$-category, we develop an efficient homotopy theory of cyclonic objects: circle-equivariant objects relative to the family of finite subgroups. We construct an $\infty$-category of cyclotomic spectra as the homotopy fixed points of an action of the multiplicative monoid of the natural numbers on the category of cyclonic spectra. Finally, we elucidate and prove a conjecture of Kaledin on cyclotomic complexes.
Given symmetric monoidal infinity-categories C and D, subject to mild hypotheses on D, we define an infinity-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). An E_infinity monoid for the Day convolution product is a lax monoidal functor from C to D.
We define a functorial spectrum-level filtration on the topological Hochschild homology of any commutative ring spectrum $R$, and more generally the factorization homology $R \otimes X$ for any space $X$, echoing algebraic constructions of Loday and Pirashvili. We investigate the properties of this filtration and show that it breaks THH up into common eigenspectra of the Adams operations.
We define a convenient $\infty$-operad parametrizing modules over commutative algebras in $\infty$-categories.
In this technical note, we proffer a very explicit construction of the "dual cocartesian fibration" $p^{\vee}$ of a cartesian fibration $p$, and we show they are classified by the same functor to $\mathbf{Cat}_{\infty}$.