Babytop Seminar

The seminar will meet Tuesdays at 3:30pm in E25-117 unless otherwise noted.

5.12.15: Sergio Estrada (MIT).
Model category structures arising from complete cotorsion pairs

In the talk we discuss the interlacing between the so-called complete cotorsion pairs and (exact) model category structures in exact categories. We will also see that exact model category structures almost always contain a Quillen equivalent Frobenius model category. We illustrate the theory by constructing, among others, a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves. The talk summarizes recent works by Hovey, Gillespie and the speaker.

2.17.15: John Harper (Ohio State).

2.10.15: Nat Stapleton (Bonn).
The character of the total power operation

We construct a total power operation on n-fold class functions
compatible with the total power operation in Morava E-theory through
the character map of Hopkins, Kuhn, and Ravenel. In essence, this
gives a formula for the total power operation in Morava E-theory
applied to a finite group. This is joint work with Barthel.

To a “stable homotopy theory” (a presentable, symmetric
monoidal stable ∞-category), we naturally associate a category of finite
e ́tale algebra objects and, using Grothendieck’s categorical machine,
a profinite group that we call the Galois group. This construction
builds on, and generalizes, ideas of many authors, and includes the e
́tale fundamental group of algebraic geometry as a special case. We
calculate the Galois groups in several examples, both in settings of
“chromatic” stable homotopy theories and modular representations of
finite groups. For instance, we show that the Galois group of the
periodic E∞ -algebra of topological modular forms is trivial, and,
extending work of Baker and Richter, that the Galois group of
K(n)-local stable homotopy theory is an extended version of the Morava
stabilizer group.

11.19.13: Aaron Mazel-Gee (Berkeley).
Every love story is a GHOsT story: Goerss--Hopkins obstruction theory for ∞-categories.

Goerss--Hopkins obstruction theory is a tool for obtaining structured ring spectra from purely algebraic data, originally conceived as the main ingredient in the construction of *tmf* as an *E*_{∞}-ring spectrum. However, while the story is extremely beautiful, it is also absurdly intricate. Part of this is because the real mathematical ideas at its core are quite deep, but a good deal of the complexity arises from an overwhelming amount of model-categorical technicalities.

In this talk, I will present a generalization of Goerss--Hopkins obstruction theory for presentable ∞-categories. At this level of abstraction, the entire story becomes...well, perhaps not tautological, but certainly a whole lot clearer. It takes a while to get there, though, and so this will be a two-part talk with a short break in the middle (for which I will provide cookies). This all may sound a bit daunting, but I give you my word as a gentleman and a topologist that you will come out with a much better understanding of Goerss--Hopkins obstruction theory than you did going in. Unless you're Mike Hopkins: then I make no promises.

Statement: Let K be any reduced cohomology theory. If a space X is K-acyclic i.e. K(X)=0 then K(P_n X)=0 for any Postnikov piece of X.
True or false? In general false of course but
Theorem: True if X is nilpotent.
This is proved by considering pointed homotopy colimits constructed out of a single space A and approximation of other spaces via such colimits. We will explain some properties of the resulting cellularization functor. The above is an immediate consequence.

The Barratt-Priddy-Quillen theorem, arguably one of the most striking in topology, states that the K-theory of the category of finite sets is equivalent to the sphere spectrum. We'll state and prove a multiplicative analog of the theorem, giving a simple category whose K-theory can be identified with the free E-infinity ring spectrum on one generator. If time permits, we'll discuss the combinatorics of divided power spectra and other future directions.

Classically, homotopy theories are described using homotopical algebra, e.g. as model categories or (co)fibration categories. Nowadays, they are often formalized as higher categories, e.g. as quasicategories or complete Segal spaces. These two types of approaches highlight different aspects of abstract homotopy theory and are useful for different purposes. Thus it is an interesting question whether homotopical algebra and higher category theory are in some precise sense equivalent.
In this talk I will concentrate on cofibration categories and quasicategories. I will discuss some basic features of both notions building up to a result that the homotopy theory of cofibration categories is indeed equivalent to the homotopy theory of cocomplete quasicategories.

We study principal bundles for strict Lie n-groups over simplicial manifolds. Given a Lie group G, one can construct a principal G-bundle on a manifold M by taking a cover U of M, specifying a transition cocycle, and then quotienting U x G by the equivalence relation generated by the cocycle. We demonstrate the existence of an analogous construction for arbitrary strict Lie n-groups. As an application, we show how our construction leads to a simple finite dimensional model of the Lie 2-group String(n).
Note: this talk will be at 4pm.

11.13.12: Luis (MIT).
The Goodwillie tower of the identity in $Alg_{\mathcal{O}}(Sp)$ (with a side helping of general context for Goodwillie Cauculus).

The main goal of this talk will be to show that the Goodwillie tower of the identity in $Alg_{\mathcal{O}}(Sp)$, where $\mathcal{O}$ is a spectral operad, is given by the truncation functors
$\mathcal{O}_{\leq n}\circ_{\mathcal{O}}$ (this tower as first by studied by Harper and Hess).
The first part of the talk will provide an overview to a more general context for Goodwillie calculus in model categories (as the treatment by Goodwillie only dealt with the cases of Top and Spec), lightly touching on some of the technical problems that arise.
Then, in the second part, we apply that context to $Alg_{\mathcal{O}}(Sp)$ and, using the fact that the stabilization of $Alg_{\mathcal{O}}$ is $Mod_{\mathcal{O}(1)}$, show that indeed the Harper-Hess tower is the Goodwillie tower.

We define $E_\infty$ ring spaces and $E_\infty$ ring spectra operadically and discuss the relation between the two categories. We then explain how to obtain $E_\infty$ ring spaces from bipermutative categories.

I will attempt to give a gentle introduction to $\infty$-categories, and introduce the notion of a monoidal $\infty$-category. Time permitting, I will then try to explain how to generalize these to get $\infty$-operads. The talk will contain at least one joke, and a large number of infinity-symbols.

You may have heard the buzz about quasi-categories, which are homotopical versions of ordinary categories. Just as the collection of ordinary categories should be arranged into a 2-category, the collection of quasi-categories should form a quasi-2-category. Continuing, the collection of quasi-2-categories should form a quasi-3-category, etc. etc. In this talk, we will define quasi-$\omega$-categories, of which quasi-$n$-categories are examples, and if time permits we will construct the quasi-$\omega$-category of quasi-$\omega$-categories.

I will talk about Mark Behren's work on the Goodwillie tower and
the EHP sequence. Calculating the homotopy groups of spheres at the prime 2 is the most important question in algebraic topology (for me). In 1937 Freudenthal proved the suspension theorem, which shows that many of the homotopy groups of spheres are stable. Using techniques such as the Adams spectral sequence, people have managed to compute the stable homotopy groups of spheres in an impressively large range. Mark shows that we can use a sound understanding of the stable groups to compute the unstable groups. In particular, there is a transfinite spectral sequence from the stable homotopy groups of spheres to the stable homotopy groups of
spheres (inception?!), which when truncated gives the unstable groups (minus resolving some extension problems). I will try my hardest to make the talk accessible to those with a fear of spectral sequences and I will recap what the EHP sequence and the Goodwillie tower are before using them!

In this talk we will discuss Rezk's proposed model for the model category of $(\infty,n)$-category types (and, more generally, $(n+k,n)$-categories), which has the particularly nice property of being cartesian closed, hence making easy the task of defining the correct $(\infty,n)$-category of maps between two $(\infty,n)$-categories.
We'll cover (and motivate) his definitions, as well as sketch the proof of cartesian closedness.

We'll go further into last week's discussion of the EHP fibre sequence,
and make some computations in the resulting spectral sequence.

We will prove the 2-primary exponent theorem of James: 4^n annihilates the
2-primary components of \pi_k(S^{2n+1}) for all k>2n+1. The tools used to
prove this result are fairly basic and this talk should be comprehensible
to anyone who took 18.906. There is an analogous result for odd primes;
different techniques are required for the even prime because S^{2n+1}
localised at 2 is not necessarily an H-space. In particular, we'll make use
of Samelson products and the Hilton-Milnor theorem, which gives a beautiful
decomposition of \Omega\Sigma(X v Y) for connected spaces X and Y.

This is a Tuesday talk of the Topology seminar.

In this talk, I'll attempt to give non-platitudinous answers to some or any of the following questions: is an "affine derived scheme" anything more than a commutative ring spectrum doing a headstand? What is algebra? What are the points of Spec S? Is the homotopy category good for anything? What are the points of Spec R? What is a ring? How much can we inflate the word Spec before it explodes anyway? etc. etc.

In 1999, Chas and Sullivan constructed a product on the homology of the free loops on an oriented manifold that looks very similar to the intersection product. Moreover they extended this product to a Gerstenhaber (and even BV) structure. This structure became much clearer with work of Cohen and Jones showing that the Hochschild cochains of the singular cochains of M were quasi-isomorphic to the chains on the Thom spectrum LM^{-TM}. Dwyer Miller and Klein extended this further by showing that LM^{-TM} was actually the topological Hochschild cohomology of the infinite suspension of based loops on M. The talk will be about Cohen and Jones result. All necessary background will be provided.

The generalized character map of HKR can be rephrased as saying
that the connected p-divisible group corresponding to the universal
formal group law over Morava E_n becomes constant over an appropriate
extension, L, of E_n. Here L is a colimit of inverted Drinfel'd rings
of level structures.
This ring L remembers E_n: it is faithfully flat over p^{-1}
E_n; furthermore, p^{-1} E_n can be recovered from L by takinginvariants with respect to a naturally occurring group action.
I will talk about this story and ask how it might be generalized to an
intermediate setting.

One remarkable property of the $E_n$ operads is their apparent self-duality: it would seem that the Koszul dual of the $E_n$ operad is a $-n$ shifted version of itself. This is first suggested by examining the corresponding homology operad, which looks like an amalgamation of the commutative operad with the ($n-1$ shifted) Lie operad. A first proof at this (homology) level was first found by Getzler and Jones, with a much more recent proof at the chain complex level having been found by Fresse. A full proof at the most general level (Spaces/Spectra) seems however not to have yet been found.
In this talk we will, after defining the relevant concepts, discuss Getzler and Jones proof, which uses interesting compact models for the $E_n$ operads known as the Fulton McPherson operads, and, time permiting (and contigent on the speakers ability to both understand and be enlightning about them), the main ideas behind Fresse's more recent result.

Sometimes a commutative ring R carries a ``logarithm'' map from the units
in R to the underlying additive group of R. In the case we'll be concerned
with, R is a K(1)-local commutative ring spectrum, and the existence of
such a logarithm follows abstractly from the Bousfield-Kuhn functor. A
priori this log is pretty opaque from a practical standpoint, but Rezk
managed to find a formula for it as an infinite series. This talk is about
Rezk's formula. Or I guess you could say it's about how Bott periodicity
manages to magically produce the p-adic logarithm and other interesting
series besides.

Crystalline cohomology is the first robust headspace in which one can understand
the p-torsion in the cohomology of a variety over a field of characteristic p,
patching a notorious puncture in the great bicycle tyre that is Weil cohomology.
I'll begin by laying out a manifesto which states what we want to achieve, and
I'll go on to sketch an achievement of it, whose vivid chapters include the
functoriality of the crystalline topos and the isomorphism with de Rham
cohomology. If permitted by (i) time and (ii) the quantity of knowledge I can
guzzle in the next twenty-four hours, I'll serve a portion of the magnificent de
Rham-Witt complex, which explicitly computes crystalline cohomology.

Taking the infinite loop spaces of the mod-2
Adams tower of a
spectrum gives a spectral sequence converging to the cohomology of the
infinite loop space of the spectrum, whose E_2-term can be identified with
certain algebraic derived functors. I'll explain how to set this up, then
say something about the computation of these derived functors.

10.25.11: Tomer Schlank (Hebrew University, Jerusalem). A Projective Model Structure on Pro-categories , and the Relative
\'Etale Homotopy Type.

Isaksen showed that a proper model category
$C$, induces a model structure
on the pro-category $Pro(C)$.
In this talk I will present a new method for defining a model structure on
the pro-category $Pro(C)$. This method requires $C$ to satisfy a much
weaker condition then having a model structure. The main application will be
a novel model structure on pro-simplicial sheaves. We see that in this
model structure a "topological lift" of Artin and Mazur's \'Etale homotopy
type is naturally obtained as an application of some natural derived functor
to the terminal object of the \'etale topos. This definition can be
naturally generalized to a relative setting, namely- given a map of topoi T
\to S, we get a notion of a relative homtopy type of T over S which is a
Pro-simplicial object in S.
This definition turns out to be useful for the study of rational points on
algebraic varieties.
This is a joint work with Ilan Barnea

In this talk we will give an overview
of THH (Topological Hochschild Homology) which is the analogue in the category
of spectra of good-old Hochschild homology for associative algebra over field.
We will give two different construction of THH. The first one through the cyclic
bar construction has the advantage of being a straightforward generalization of
the algebraic version. The second one through factorization homology is more
interesting as it describes THH as an example inside a large family of
constructions indexed by framed manifolds.
Finally if time permits we will introduce the Bockstedt spectral sequence and
make an explicit computation of THH(KU).

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