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Clark Barwick

Publications and source records attributed to Clark Barwick.

At least 19 recordsLinked to original sources
arXiv2026

Factorization algebras in quite a lot of generality

This is a first stab at a mathematical framework in which one can study quantum field theories on spacetimes with quite general geometries. We will study these theories via their factorization algebras. The aim is to identify a minimalist formalism that makes sense of factorization algebras in any geometric context. This formalism extends the technology of factorization algebras to many new contexts, including those arising in arithmetic quantum field theories. In order to make sense of factorization algebras on a geometric object X, one needs two ingredients. First, one needs an additional piece of structure on X that we call an "isolability structure." This is the data required to say whether two (generalized) points of X are "distant." This is encoded as a functor from a certain combinatorial category of cographs. Second, one needs some sort of sheaf theory. The isolability structure then induces on the category of sheaves a kind of twofold symmetric monoidal structure. Factorization algebras are then defined in terms of this structure. This paper develops this formalism. We describe how some existing theories of factorization algebras fit into this framework, and we give a construction of the Beilinson-Drinfeld Grassmannian as a factorization stack that works in quite a lot of generality.

math.QA↗
arXiv2021

K-theory and polynomial functors

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 B\"okstedt's calculation of $\mathrm{THH}(\mathbb{F}_p)$.

math.KT↗
arXiv2019

Pyknotic objects, I. Basic notions

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 derived category of a local field. In this article, we present the basic theory of pyknotic objects, with a view to describing a simple set of everyday examples.

math.AG↗
arXiv2019

Exodromy for stacks

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 the Galois category of a simplicial scheme is equivalent to its \'etale topological type in the sense of Friedlander.

math.AG↗
arXiv2018

A comment on the vanishing of rational motivic Borel-Moore homology

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 that this conjecture holds if and only if the cyclic action on the motivic cohomology of an Artin--Schreier field extension in bidegree $(i,j)$ is trivial if $i<j$.

math.AG↗
arXiv2018

On Galois categories and perfectly reduced schemes

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 (or morphism of such) and the categorical properties of its Galois category (or functor of such). The main thing that makes this possible is Schr\"oer's total separable closure.

math.AG↗
arXiv2018

Exodromy

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 \'etale 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 \'etale 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 \'etale homotopy type of $X$. We also prove a higher categorical form of Hochster Duality, which reconstructs the entire \'etale 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.

math.AT↗
arXiv2016

Categorifying rationalization

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, $K_n(S^{-1}E)\cong S^{-1}K_n(E)$.

math.KT↗
arXiv2016

Parametrized higher category theory and higher algebra: Expos\'e I -- Elements of parametrized higher category theory

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 Expos\'e 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.

math.AT↗
arXiv2016

Fibrations in $\infty$-category theory

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 category theory.

math.CT↗
arXiv2016

On the fibrewise effective Burnside $\infty$-category

Effective Burnside $\infty$-categories are the centerpiece of the $\infty$-categorical approach to equivariant stable homotopy theory. In this \'etude, 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.

math.CT↗
arXiv2016

A note on stable recollements

In this short \'etude, 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.

math.CT↗
arXiv2016

Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin

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.

math.AT↗
arXiv2014

Dualizing cartesian and cocartesian fibrations

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 $\mathbf{Cat}_{\infty}$.

math.CT↗
arXiv2011

On the Unicity of the Homotopy Theory of Higher Categories

We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $\Theta_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, the first author's $n$-fold complete Segal spaces, Kan and the first author's $n$-relative categories, and complete Segal space objects in any model of $(\infty,n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb{Z}/2)^n$.

math.AT↗
arXiv2007

On (Enriched) Left Bousfield Localization of Model Categories

I verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and I prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. I also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition.

math.AT↗