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Jonathan Beardsley

Publications and source records attributed to Jonathan Beardsley.

18 recordsLinked to original sources
arXiv2025

The Eilenberg-MacLane Spectrum of \mathbb{F}_1

Given a very special $\Gamma$-space $X$, repeated application of Segal's delooping functor produces the constituent spaces of the associated connective $\Omega$-spectrum. In particular, by applying this construction to \textit{discrete} very special $\Gamma$-spaces (a.k.a.~Abelian groups), one recovers Eilenberg-MacLane spectra. The delooping functor is entirely formal, however, and can be applied to arbitrary $\Gamma$-spaces without any conditions. Work of Connes and Consani suggests that the ``field with one element'' can be fruitfully realized as a (discrete) $\Gamma$-space (which localizes to the classical sphere spectrum). This note computes Segal's deloopings of this model of $\mathbb{F}_1$. They are $n$-fold simplicial sets whose geometric realizations are the $n$-spheres, equipped with \textit{free partial commutative monoid} structures. Equivalently, they are the (nerves of the) free partial strict $n$-categories with free partial symmetric monoidal structures.

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arXiv2025

Dynkin Systems and the One-Point Geometry

In this note I demonstrate that the collection of Dynkin systems on finite sets assembles into a Connes-Consani $\mathbb{F}_1$-module, with the collection of partitions of finite sets as a sub-module. The underlying simplicial set of this $\mathbb{F}_1$-module is shown to be isomorphic to the delooping of the Krasner hyperfield $\mathbb{K}$, where $1+1=\{0,1\}$. The face and degeneracy maps of the underlying simplicial set of the $\mathbb{F}_1$-module of partitions correspond to merging partition blocks and introducing singleton blocks, respectively. I also show that the $\mathbb{F}_1$-module of partitions cannot correspond to a set with a binary operation (even partially defined or multivalued) under the ``Eilenberg-MacLane'' embedding. These results imply that the $n$-fold sum of the Dynkin $\mathbb{F}_1$-module with itself is isomorphic to the $\mathbb{F}_1$-module of the discrete projective geometry on $n$ points.

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arXiv2024

Higher Groups and Higher Normality

In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary $\infty$-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a "normal closure" monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of $\infty$-topoi preserve normal maps or, equivalently, that monoidal functors of $\infty$-topoi preserve the property of "being a fiber" for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first two of them, and provide counterexamples to the third. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in $\infty$-topoi in the "usual way." One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in $\infty$-topoi.

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arXiv2024

Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules

We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\fun$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some time elaborating its structure.

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arXiv2023

Brauer-Wall Groups and Truncated Picard Spectra of $K$-theory

We compute the first two k-invariants of the Picard spectra of $KU$ and $KO$ by analyzing their Picard groupoids and constructing their unit spectra as global sections of sheaves on the category of manifolds. This allows us to determine the E_\infty-structures of their truncations Pic(KU)[0,3] and Pic(KO)[0,2]. It follows that these truncated Picard spaces represent: the Brauer groups of Z/2-graded algebra bundles of Donovan-Karoubi, Moutuou and Maycock; the Brauer groups of super 2-lines; and the K-theory twists of Freed, Hopkins and Teleman. Our results also imply that that these spaces represent twists of String and Spin structures on manifolds and can be used to twist tmf-cohomology. Finally, we are able to identify pic(KU)[0,3] with a cotruncation of the Anderson dual of the sphere spectrum.

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arXiv2022

Labelled cospan categories and properads

We prove Steinebrunner's conjecture on the biequivalence between (colored) properads and labelled cospan categories. The main part of the work is to establish a 1-categorical, strict version of the conjecture, showing that the category of properads is equivalent to a category of strict labelled cospan categories via the symmetric monoidal envelope functor.

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arXiv2021

Skeleta and categories of algebras

We define a notion of a connectivity structure on an $\infty$-category, analogous to a $t$-structure but applicable in unstable contexts -- such as spaces, or algebras over an operad. This allows us to generalize notions of n-skeleta, minimal skeleta, and cellular approximation from the category of spaces. For modules over an Eilenberg-Mac Lane spectrum, these are closely related to the notion of projective amplitude. We apply these to ring spectra, where they can be detected via the cotangent complex and higher Hochschild homology with coefficients. We show that the spectra $Y(n)$ of chromatic homotopy theory are minimal skeleta for $H\mathbb{F}_2$ in the category of associative ring spectra. Similarly, Ravenel's spectra $T(n)$ are shown to be minimal skeleta for $BP$ in the same way, which proves that these admit canonical associative algebra structures.

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arXiv2019

Koszul Duality in Higher Topoi

We show that there is an equivalence in any $n$-topos $\mathcal{X}$ between the pointed and $k$-connective objects of $\mathcal{X}$ and the $\mathbb{E}_k$-group objects of the $(n-k-1)$-truncation of $\mathcal{X}$. This recovers, up to equivalence of $\infty$-categories, some classical results regarding algebraic models for $k$-connective, $(n-1)$-coconnective homotopy types. Further, it extends those results to the case of sheaves of such homotopy types. We also show that for any pointed and $k$-connective object $X$ of $\mathcal{X}$ there is an equivalence between the $\infty$-category of modules in $\mathcal{X}$ over the associative algebra $\Omega^k X$, and the $\infty$-category of comodules in $\mathcal{X}$ for the cocommutative coalgebra $\Omega^{k-1}X$. All of these equivalences are given by truncations of Lurie's $\infty$-categorical bar and cobar constructions, hence the terminology "Koszul duality".

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arXiv2018

On Bialgebras, Comodules, Descent Data and Thom Spectra in $\infty$-categories

This paper lays some of the foundations for working with not-necessarily-commutative bialgebras and their categories of comodules in $\infty$-categories. We prove that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category (as opposed to a relative (co)tensor product over the underlying algebra or coalgebra of the bialgebra). We give two examples of higher coalgebraic structure: first, following Hess we show that for a map of $\mathbb{E}_n$-ring spectra $\phi\colon A\to B$, the associated $\infty$-category of descent data is equivalent to the category of comodules over $B\otimes_A B$, the so-called descent coring; secondly, we show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the $\infty$-categorical Thom diagonal of Ando, Blumberg, Gepner, Hopkins and Rezk (which we describe explicitly) and that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way.

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arXiv2018

The Operadic Nerve, Relative Nerve, and the Grothendieck Construction

We relate the relative nerve $\mathrm{N}_f(\mathcal{D})$ of a diagram of simplicial sets $f \colon \mathcal{D} \to \mathsf{sSet}$ with the Grothendieck construction $\mathsf{Gr} F$ of a simplicial functor $F \colon \mathcal{D} \to \mathsf{sCat}$ in the case where $f = \mathrm{N} F$. We further show that any strict monoidal simplicial category $\mathcal{C}$ gives rise to a functor $\mathcal{C}^\bullet \colon \Delta^\mathrm{op} \to \mathsf{sCat}$, and that the relative nerve of $\mathrm{N} \mathcal{C}^\bullet$ is the operadic nerve $\mathrm{N}^\otimes(\mathcal{C})$. Finally, we show that all the above constructions commute with appropriately defined opposite functors.

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arXiv2018

The Enriched Grothendieck Construction

We define and study opfibrations of $V$-enriched categories when $V$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category $B$, there is an equivalence of 2-categories between $V$-enriched opfibrations over the free $V$-category on $B$, and pseudofunctors from $B$ to the 2-category of $V$-categories. This generalizes the classical ($Set$-enriched) Grothendieck correspondence.

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arXiv2017

Toward a Galois theory of the integers over the sphere spectrum

Recent work in higher algebra allows the reinterpretation of a classical description of the Eilenberg-MacLane spectrum $H\mathbb{Z}$ as a Thom spectrum, in terms of a kind of derived Galois theory. This essentially expository talk summarizes some of this work, and suggests an interpretation in terms of configuration spaces and monoidal functors on them, with some analogies to a topological field theory.

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arXiv2017

Topological Hochschild homology of X(n)

We show that Ravenel's spectrum $X(2)$ is the versal $E_1$-$S$-algebra of characteristic $\eta$. This implies that every $E_1$-$S$-algebra $R$ of characteristic $\eta$ admits an $E_1$-ring map $X(2)\to R$, i.e. an $\mathbb{A}_\infty$ complex orientation of degree 2. This implies that $R^\ast(\mathbb{C}P^2)\cong R_\ast[x]/x^3$. Additionally, if $R$ is an $\mathbb{E}_2$-ring Thom spectrum admitting a map (of homotopy ring spectra) from $X(2)$, e.g. $X(n)$, its topological Hochschild homology has a simple description.

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arXiv2017

A Theorem on Multiplicative Cell Attachments with an Application to Ravenel's X(n) Spectra

We show that the homotopy groups of a connective $E_k$-ring spectrum with an $E_k$-cell attached along a class $\alpha$ in degree $n$ are isomorphic to the homotopy groups of the cofiber of the self-map associated to $\alpha$ through degree $2n$. Using this, we prove that the $2n-1^{st}$ homotopy groups of Ravenel's $X(n)$ spectra are cyclic for all $n$. This further implies that, after localizing at a prime, $X(n+1)$ is homotopically unique as the $E_1$-$X(n)$-algebra with homotopy groups in degree $2n-1$ killed by an $E_1$-cell. Lastly, we prove analogous theorems for a sequence of $E_k$-ring Thom spectra, for each odd $k$, which are formally similar to Ravenel's $X(n)$ spectra and whose colimit is also $MU$.

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arXiv2016

Relative Thom Spectra Via Operadic Kan Extensions

We show that a large number of Thom spectra, i.e. colimits of morphisms $BG\to BGL_1(\mathbb{S})$, can be obtained as iterated Thom spectra, i.e. colimits of morphisms $BG\to BGL_1(Mf)$ for some Thom spectrum $Mf$. This leads to a number of new relative Thom isomorphisms, e.g. $MU[6,\infty)\wedge_{MString} MU[6,\infty)\simeq MU[6,\infty)\wedge\mathbb{S}[B^3Spin]$. As an example of interest to chromatic homotopy theorists, we also show that Ravenel's $X(n)$ filtration of $MU$ is a tower of intermediate Thom spectra determined by a natural filtration of $BU$ by sub-bialagebras.

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arXiv2015

Some Higher Coalgebra

We define quasicategories of E_n-structured coalgebras, bialagebras and comodules. We show that: n-fold loop spaces, suspension spectra thereof, descent data for maps of E_n-ring spectra, descent corings of morphisms of E_n-ring spectra and Thom spectra are all examples of these kinds of objects. In particular, we prove that for a morphism of E_n-monoidal Kan complexes f:X->BGL_1(R), the associated Thom spectrum Mf is a structured R[X]-comodule by the classical Thom diagonal.

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arXiv2011

A Sheaf of Boehmians

We show that Boehmians defined over open sets of $\mathbb{R}^N$ constitute a sheaf. In particular, it is shown that such Boehmians satisfy the gluing property of sheaves over a topological space.

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