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.
In this note I demonstrate that the collection of Dynkin systems on finite sets assembles into a Connes-Consani π½β-module, with the collection of partitions of finite sets as a sub-module. The underlying simplicial set of this π½β-module is shown to be isomorphic to the delooping of the Krasner hyperfield π, where 1+1={0,1}. The face and degeneracy maps of the underlying simplicial set of the π½β-module of partitions correspond to merging partition blocks and introducing singleton blocks, respectively. I also show that the π½β-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 π½β-module with itself is isomorphic to the π½β-module of the discrete projective geometry on n points.
In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary β-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 β-topoi preserve normal maps or, equivalently, that monoidal functors of β-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 β-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 β-topoi.
We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into π½β-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 π½β-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.
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 Eβ-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.
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.
We define a notion of a connectivity structure on an β-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 Hπ½β 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.
This paper lays some of the foundations for working with not-necessarily-commutative bialgebras and their categories of comodules in β-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 πΌβ-ring spectra Ο:AβB, the associated β-category of descent data is equivalent to the category of comodules over BβB, 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 β-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.
We show that there is an equivalence in any n-topos π³ between the pointed and k-connective objects of π³ and the πΌβ-group objects of the (nβkβ1)-truncation of π³. This recovers, up to equivalence of β-categories, some classical results regarding algebraic models for k-connective, (nβ1)-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 π³ there is an equivalence between the β-category of modules in π³ over the associative algebra Ξ©α΅X, and the β-category of comodules in π³ for the cocommutative coalgebra Ξ©α΅β»ΒΉX. All of these equivalences are given by truncations of Lurie's β-categorical bar and cobar constructions, hence the terminology "Koszul duality".
We relate the relative nerve N(π) of a diagram of simplicial sets f:πβππ²πΎπ with the Grothendieck construction π¦πF of a simplicial functor F:πβππ’πΊπ in the case where f=NF. We further show that any strict monoidal simplicial category π gives rise to a functor Bπ:ΞΒ°βππ’πΊπ, and that the relative nerve of NBπ is the operadic nerve N^β(π). Finally, we show that all the above constructions commute with appropriately defined opposite functors.
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.
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.
We show that the homotopy groups of a connective πΌβ-ring spectrum with an πΌβ-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 (2nβ1)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 πΌβ-X(n)-algebra with homotopy groups in degree 2nβ1 killed by an πΌβ-cell. Lastly, we prove analogous theorems for a sequence of πΌβ-ring Thom spectra, for each odd k, which are formally similar to Ravenel's X(n) spectra and whose colimit is also MU.
We show that a large number of Thom spectra, i.e. colimits of morphisms BGβBGLβ(π), can be obtained as iterated Thom spectra, i.e. colimits of morphisms BGβBGLβ(Mf) for some Thom spectrum Mf. This leads to a number of new relative Thom isomorphisms, e.g. MU[6,β)β§_{MString}MU[6,β)βMU[6,β)β§π[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.