Chapter from a forthcoming book on spectra, surveying some basic ideas about spectral algebraic geometry.
We present a calculation, which shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of U(1)-bundles on a torus. More generally, we show how the analogous calculation, applied to a moduli space of principal bundles for a K(Z,2) central extension of U(1)d$ give rise to Looijenga line bundles. We then speculate on the relation of these calculations to complex analytic equivariant elliptic cohomology.
We give a survey of a generalization of Quillen-Sullivan rational homotopy theory which gives spectral algebra models of unstable v_n-periodic homotopy types. In addition to describing and contextualizing our original approach, we sketch two other recent approaches which are of a more conceptual nature, due to Arone-Ching and Heuts. In the process, we also survey many relevant concepts which arise in the study of spectral algebra over operads, including topological Andr\'e-Quillen cohomology, Koszul duality, and Goodwillie calculus.
A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of Map(BG,BH) and (BGH)G, showing that they are computed entirely in terms of spaces of homomorphisms from G to H. These results generalize the well-known case when H is finite, and the case of H compact abelian due to Lashof, May, and Segal.
The modern understanding of the homotopy theory of spaces and spectra is organized by the chromatic philosophy, which relates phenomena in homotopy theory with the moduli of one-dimensional formal groups. In this paper, we describe how certain phenomena of K(n)-local homotopy theory can be computed from knowledge of isogenies of deformations of formal groups of height n
Abstract: We define a notion of ``Frobenius pair'', which is a mild generalization of the notion of ``Frobenius object'' in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold forms a ``commutative Frobenius pair'' in the stable homotopy category of spectra.
We show that the ring of power operations for any Morava E-theory is Koszul.
While many different models for (∞,1)-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for (∞, n)-categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as (∞,n)-categories. Furthermore, we establish Quillen equivalences between them.
We use the notion of multi-Reedy category to prove that, if C is a Reedy category, then ΘC is also a Reedy category. This result gives a new proof that the categories Θn are Reedy categories. We then define elegant Reedy categories, for which we prove that the Reedy and injective model structures coincide.
We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the string orientation of tmf, the spectrum of topological modular forms. We also develop the analogous theory of Thom spectra and orientations for associative ring spectra. Our work is based on a new model of the Thom spectrum as a derived smash product. An earlier version of this paper was part of arXiv:0810.4535.
We develop a generalization of the theory of Thom spectra using the language of infinity categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parametrized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an associative ring spectrum R, we associate a Thom spectrum to a map of infinity categories from the infinity groupoid of a space X to the infinity category of free rank one R-modules, which we show is a model for BGL1R$; we show that $BGL1R$ classifies homotopy sheaves of rank one R-modules, which we call R-line bundles. We use our R-module Thom spectrum to define the twisted R-homology and cohomology of an R-line bundle over a space X, classified by a map from X to $BGL1R, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory. An earlier version of this paper was part of arXiv:0810.4535.
Abstract: We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2.
Abstract: We give a shorter proof of Lemma 1.9 from Goodwillie, "Calculus III", which is the key step in proving that the construction PnF gives an n-excisive functor.
We prove a congruence criterion for the algebraic theory of power operations in Morava E-theory, analogous to Wilkerson's congruence criterion for torsion free λ-rings. In addition, we provide a geometric description of this congruence criterion, in terms of sheaves on the moduli problem of deformations of formal groups together with deformations of Frobenius isogenies.
We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Θ-spaces. The (n+k,n)-Θ-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Θn. This notion is a generalization of that of complete Segal spaces (which are precisely the (∞,1)-Θ-spaces). Our main result is that the above model category is cartesian.
We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the "building complex" associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF(Gamma0(3)).
We construct a "logarithmic" cohomology operation on Morava E-theory, which is a homomorphism defined on the multiplicative group of invertible elements in the ring E0(K) of a space K. We obtain a formula for this map in terms of the action of Hecke operators on Morava E-theory. Our formula is closely related to that for an Euler factor of the Hecke L-function of an automorphic form.
We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum LK(2)S0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E2hF where F is a finite subgroup of the Morava stabilizer group and E2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of the fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic.
Abstract: We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.
Abstract: We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory, which is allowed to be multi-sorted. The results have applications to the construction of localization model category structures.
Abstract: We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models has a well-behaved internal hom-object.
Abstract: We show that the class of p-complete connective spectra with finitely presented cohomology over the Steenrod algebra admits a duality theory related to Brown-Comenetz duality. This construction also produces a full-plane version of the classical Adams spectral sequence for such spectra, which converges to the homotopy groups of a ``finite'' localization.
Abstract: We give an exposition of the proof of a theorem of Hopkins and Miller, that the spectra En admit an action of the Morava stabilizer group.
We describe a generalization of κ-filtered ∞-category, where the role regular cardinal κ is replaced by a class of small ∞-categories. This leads to a possible generalization of the notion of accessible ∞-category.
These are notes for my lectures in a workshop on "Higher topos theory and univalent foundations", Leeds, June 2019.
We prove the following fact which is difficult to find in the literature: if p: X → S is a Cartesian fibration of arbitrary simplicial sets, and f is a degenerate edge of X, then f is a p-Cartesian edge.
A summary of basic facts about the "standard" category of compactly generated spaces introduced by McCord.
Lecture notes for a series of talks given in Bonn, June 2015. Most of the topics covered touched in one way or another on the role of power operations in elliptic cohomology.
An exposition of some proofs of the Freudenthal suspension theorem and the Blakers-Massey theorem. These are meant to be reverse engineered versions of proofs in homotopy type theory due to Lumsdaine, Finster, and Licata. The proof of Blakers-Massey given here is based on a formalization given by Favonia.
Given a λ-ring A and a formally etale morphism f: A → B of commutative rings, one may ask: What are the possible λ-ring strutures on B such that f is a map of λ-rings? We give the answer: Such a lifted λ-ring structure on B is determined uniquely by a compatible choice of lifts of the Adams operations ψp from A to B for all primes p which satisfy Frobenius congruences. In other words, to extend a λ-ring structure along a formally etale morphism, we need not be concerned about the "non-linear" part of the λ-ring structures in question.
We investigate the role of cohesion in unstable global equivariant homotopy theory. (Updated 10 Mar 2014.)
The goal of this note is to understand how to prove things about geometric realizations of pullbacks, without using the dreaded "π*-Kan condition".
This is an expository treatment of what we call ``analytic completion'' of R-modules, which is a kind of completion defined in terms of quotients of power series modules. It is closely related to the left derived functors of adic completion, in which guise it has been studied by several authors.
We review what is known about power operations for Morava E-theory, and carry out some sample calculations in the height 2 case.
We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b -> bgl(A), we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b -> bgl(A) -> bgl(R) is null.
If A is an associative ring spectrum, then to a map of spaces f: B -> BGL(A) we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -> BGL(A) -> BGL(R) is null. We also note that BGL(A) classifies the twists of A-theory.
We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.
Abstract: Explicit calculations of the algebraic theory of power operations for a specific Morava E-theory spectrum are given, without detailed proofs.
Abstract: We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and seems to be of independent interest.
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