Clark Barwick
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View article: K-theory and polynomial functors
K-theory and polynomial functors Open
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 $\mathrm{THH}(\mathbb{F}_p)$.
View article: Topological Feature Extraction and Visualization of Whole Slide Images using Graph Neural Networks
Topological Feature Extraction and Visualization of Whole Slide Images using Graph Neural Networks Open
Whole-slide images (WSI) are digitized representations of thin sections of stained tissue from various patient sources (biopsy, resection, exfoliation, fluid) and often exceed 100,000 pixels in any given spatial dimension. Deep learning ap…
View article: Topological Feature Extraction and Visualization of Whole Slide Images using Graph Neural Networks
Topological Feature Extraction and Visualization of Whole Slide Images using Graph Neural Networks Open
Whole-slide images (WSI) are digitized representations of thin sections of stained tissue from various patient sources (biopsy, resection, exfoliation, fluid) and often exceed 100,000 pixels in any given spatial dimension. Deep learning ap…
View article: Pyknotic objects, I. Basic notions
Pyknotic objects, I. Basic notions Open
Pyknotic objects are (hyper)sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derive…
View article: Exodromy for stacks
Exodromy for stacks Open
In this short note we extend the Exodromy Theorem of arXiv:1807.03281 to a large class of stacks and higher stacks. We accomplish this by extending the Galois category construction to simplicial schemes. We also deduce that the nerve of th…
View article: CATEGORIFYING RATIONALIZATION
CATEGORIFYING RATIONALIZATION Open
We construct, 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 $\infty$ …
View article: A comment on the vanishing of rational motivic Borel-Moore homology
A comment on the vanishing of rational motivic Borel-Moore homology Open
This note concerns a weak form of Parshin's conjecture, which states that the rational motivic Borel--Moore homology of a quasiprojective variety of dimension $m$ over a finite field in bidegree $(s,t)$ vanishes for $s>m+t$. It is shown th…
View article: On Galois categories and perfectly reduced schemes
On Galois categories and perfectly reduced schemes Open
It turns out that one can read off facts about schemes up to universal homeomorphism from their Galois categories. Here we propose a first modest slate of entries in a dictionary between the geometric features of a perfectly reduced scheme…
View article: Exodromy
Exodromy Open
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…
View article: Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory
Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory Open
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 …
View article: Parametrized higher category theory and higher algebra: A general introduction
Parametrized higher category theory and higher algebra: A general introduction Open
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.
View article: Fibrations in $\infty$-category theory
Fibrations in $\infty$-category theory Open
In this short expository note, we discuss, with plenty of examples, the bestiary of fibrations in quasicategory theory. We underscore the simplicity and clarity of the constructions these fibrations make available to end-users of higher ca…
View article: On the fibrewise effective Burnside $\infty$-category
On the fibrewise effective Burnside $\infty$-category Open
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 proo…
View article: A note on stable recollements
A note on stable recollements Open
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…
View article: Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin
Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin Open
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 spec…
View article: On the algebraic<i>K</i>-theory of higher categories
On the algebraic<i>K</i>-theory of higher categories Open
We prove that Waldhausen $K$-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, $K$-theory spaces admit canonical (connective) deloopings, and the $K$-theory func…
On exact -categories and the Theorem of the Heart Open
The new homotopy theory of exact $\infty$ - categories is introduced and employed to prove a Theorem of the Heart for algebraic $K$ -theory (in the sense of Waldhausen). This implies a new compatibility between Waldhausen $K$ -theory and N…
View article: Multiplicative structures on algebraic $K$-theory
Multiplicative structures on algebraic $K$-theory Open
The algebraic K -theory of Waldhausen \infty -categories is the functor corepresented by the unit object for a natural symmetric monoidal structure. We therefore regard it as the stable homotopy theory of homotopy theories. In particular, …