Jonathan Beardsley

Research     Teaching     Other

I am broadly interested in (higher) category theory and homotopy theory, with applications to topology, algebra and combinatorics. My early work was focused on chromatic homotopy theory and, in particular, providing a new way to understand Devinatz, Hopkins and Smith's proof of Ravenel's Nilpotence Conjecture. More recently, I have become interested in understanding Connes and Consani's approach to doing algebra over the "field with one element." All of my papers are available on the arXiv here.

Research Papers

Dynkin Systems and the One-Point Geometry. Submitted (arXiv).

In this note I demonstrate that the collection of Dynkin systems on finite sets assembles into a Connes-Consani 𝔽₁-module, with the collection of partitions of finite sets as a sub-module. The underlying simplicial set of this 𝔽₁-module is shown to be isomorphic to the delooping of the Krasner hyperfield 𝕂, where 1+1={0,1}. The face and degeneracy maps of the underlying simplicial set of the 𝔽₁-module of partitions correspond to merging partition blocks and introducing singleton blocks, respectively. I also show that the 𝔽₁-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 𝔽₁-module with itself is isomorphic to the 𝔽₁-module of the discrete projective geometry on n points.

Higher Groups and Higher Normality, with Landon Fox. Submitted (arXiv).

In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary ∞-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 ∞-topoi preserve normal maps or, equivalently, that monoidal functors of ∞-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 of them, and provide counterexamples to the other two. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in ∞-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 ∞-topoi.

Projective Geometries and Simple Pointed Matroids as 𝔽₁-modules, with So Nakamura. Submitted (arXiv).

We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into 𝔽₁-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 𝔽₁-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.

Brauer Wall Groups and Truncated Picard Spectra of K-theory, with Kiran Luecke and Jack Morava. Submitted (arXiv).

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∞-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 β„€/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.

Labeled Cospan Categories and Properads, with Philip Hackney. Published in Journal of Pure and Applied Algebra (arXiv).

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.

Skeleta and categories of algebras, with Tyler Lawson. Published in Advances in Mathematics (arXiv).

We define a notion of a connectivity structure on an ∞-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 H𝔽₂ 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.

On Bialgebras, Comodules, Descent Data and Thom Spectra in ∞-categories. Published in Homology, Homotopy and Applications (arXiv).

This paper lays some of the foundations for working with not-necessarily-commutative bialgebras and their categories of comodules in ∞-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 𝔼ₙ-ring spectra Ο•:Aβ†’B, the associated ∞-category of descent data is equivalent to the category of comodules over BβŠ—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 ∞-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.

Koszul duality in Higher Topoi, with Maximilien PΓ©roux. Published in Homology, Homotopy and Applications (arXiv).

We show that there is an equivalence in any n-topos 𝒳 between the pointed and k-connective objects of 𝒳 and the 𝔼ₖ-group objects of the (nβˆ’kβˆ’1)-truncation of 𝒳. This recovers, up to equivalence of ∞-categories, some classical results regarding algebraic models for k-connective, (nβˆ’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 𝒳 there is an equivalence between the ∞-category of modules in 𝒳 over the associative algebra ΩᡏX, and the ∞-category of comodules in 𝒳 for the cocommutative coalgebra Ωᡏ⁻¹X. All of these equivalences are given by truncations of Lurie's ∞-categorical bar and cobar constructions, hence the terminology "Koszul duality".

The Operadic Nerve, Relative Nerve, and the Grothendieck Construction, with Liang Ze Wong. Published in Theory and Applications of Categories (arXiv).

We relate the relative nerve N(π’Ÿ) of a diagram of simplicial sets f:π’Ÿβ†’π—Œπ–²π–Ύπ— with the Grothendieck construction 𝖦𝗋F of a simplicial functor F:π’Ÿβ†’π—Œπ–’π–Ίπ— in the case where f=NF. We further show that any strict monoidal simplicial category 𝓒 gives rise to a functor B𝓒:Ξ”Β°β†’π—Œπ–’π–Ίπ—, and that the relative nerve of NB𝓒 is the operadic nerve N^βŠ—(𝓒). Finally, we show that all the above constructions commute with appropriately defined opposite functors.

The Enriched Grothendieck Construction, with Liang Ze Wong. Published in Advances in Mathematics (arXiv).

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.

Toward a Galois Theory of the Integers Over the Sphere Spectrum, with Jack Morava. Published in Journal of Geometry and Physics (arXiv).

Recent work in higher algebra allows the reinterpretation of a classical description of the Eilenberg-MacLane spectrum Hβ„€ 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.

A Theorem on Multiplicative Cell Attachments with an Application to Ravenel's X(n) Spectra. Published in Journal of Homotopy and Related Structures (arXiv).

We show that the homotopy groups of a connective 𝔼ₖ-ring spectrum with an 𝔼ₖ-cell attached along a class Ξ± in degree n are isomorphic to the homotopy groups of the cofiber of the self-map associated to Ξ± through degree 2n. Using this, we prove that the (2nβˆ’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 𝔼₁-X(n)-algebra with homotopy groups in degree 2nβˆ’1 killed by an 𝔼₁-cell. Lastly, we prove analogous theorems for a sequence of 𝔼ₖ-ring Thom spectra, for each odd k, which are formally similar to Ravenel's X(n) spectra and whose colimit is also MU.

Relative Thom Spectra Via Operadic Kan Extensions. Published in Algebraic and Geometric Topology (arXiv).
  • A Sheaf of Boehmians, with Piotr Mikusinski. Published in Annales Polonici Mathematici (arXiv).
  • We show that a large number of Thom spectra, i.e. colimits of morphisms BGβ†’BGL₁(π•Š), can be obtained as iterated Thom spectra, i.e. colimits of morphisms BGβ†’BGL₁(Mf) for some Thom spectrum Mf. This leads to a number of new relative Thom isomorphisms, e.g. MU[6,∞)∧_{MString}MU[6,∞)≃MU[6,∞)βˆ§π•Š[BΒ³Spin]. 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.

    Other Writing

  • Matroids as 𝔽₁-modules Notes from a talk given in the UC-Irvine Algebra Seminar.
  • Group Theory for ∞-Groups Notes from a talk given in the UIUC Topology Seminar.
  • Picard Spaces and Orientations: Notes from an expository talk on Picard spaces given at the ``Chromatic Nullstellensatz Seminar."
  • Toward Higher Algebra over 𝔽₁: Notes from a talk at the conference "Low Dimensional Topology and Number Theory" at Kyushu University.
  • Interpretations of the Truncated Picard Spectra of KO and KU: Notes from a talk given at the BIRS workshop "Cobordisms, Strings and Thom Spectra," held at the Casa MatΓ©matica Oaxaca.
  • On Braids and Cobordism Theories: Notes from a talk given in University of Glasgow's topology seminar.
  • A User's Guide: Relative Thom Spectra via Operadic Kan Extensions: An exposition of the main ideas in my paper "Relative Thom Spectra via Operadic Kan Extensions," accessible to graduate students studying homotopy theory.
  • Notes on Lubin-Tate Cohomology: Some notes about the cohomology of a complex that comes up in deformations of formal groups as well as extensions of n-buds.
  • THH of X(n): A computation of the Topological Hochschild Homology of Ravenel's X(n) spectra.
  • The Harmonic Bousfield Lattice: A computation of the Bousfield lattice of the category of p-local harmonic spectra. The main theorem and proof were used by Luke Wolcott here.

    Talk Slides

  • Some Galois Theory for Bordism Homology given in the UNR Mathematics and Statistics Colloquium.
  • On the PROB of Singular Braids given at The Third Conference on Operad Theory and Related Topics.
  • Symmetry, Topology and the Nobel Prize, slides for an expository talk on topological phases of matter.
  • Twisted Forms in Homotopy Theory.
  • Bialgebras in Spectra.