skd
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Spectral algebraic geometry vs derived algebraic geometry in positive characteristic?
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26 votes

I'll try to answer this question from the topological viewpoint. The short summary is that structured objects in the spectral setting have cohomology operations and power operations, which forces ...

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Has anyone seen a nice map of multiplicative cohomology theories?
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24 votes

I'm not sure I understand what "the" map is here, but I'll attempt to answer the questions that were asked in the body of the question. Sorry if I'm just saying things that you already know. $\...

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Unifying "cohomology groups classify extensions" theorems
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16 votes

$\newcommand{\cA}{\mathcal{A}}\newcommand{\Ext}{\mathrm{Ext}}\newcommand{\Hom}{\mathrm{Hom}}$Let $\cA$ be an abelian category; then, $\Ext_\cA^i(A,B)$ is literally $\Hom_{D(\cA)}(A, B[i])$, where $B[i]...

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Homology of spectra vs homology of infinite loop spaces
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16 votes

$\newcommand{\H}{\mathrm{H}} \newcommand{\Z}{\mathbf{Z}}$Let $X$ be a space. Then the $E$-(co)homology of $X$ is the same as the $E$-(co)homology of its suspension spectrum, i.e., $E_\ast(X) \cong E_\...

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The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum
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16 votes

One can prove that $\mathrm{Map}(H\mathbf{F}_p,MU)$ is contractible. We know that $H\mathbf{F}_p$ is dissonant (Theorem 4.7 of Ravenel's "Localization with Respect to Certain Periodic Homology ...

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Spectra with "finite" homology and homotopy
16 votes

Here are two ways of thinking about it. The first comes from the way one proves the final statement you cited: if $X$ has finitely many nonzero homotopy groups which are all finitely generated, then ...

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Sphere spectrum, Character dual and Anderson dual
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16 votes

The Anderson dualizing spectrum $I_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded ...

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Natural examples of $(\infty,n)$-categories for large $n$
13 votes

$\newcommand{\Vect}{\mathrm{Vect}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\cc}{\mathbf{C}}$Here are three (related) examples. The first one is simple (although not really related to physics): an $...

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Importance of the principal bundle in Chern-Simons theory
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11 votes

In quantum Chern-Simons theory with gauge group $G$ (compact Lie), a field on a 3-manifold $M$ is a principal $G$-bundle with a connection $A$. The partition function/path integral associated to $M$ ...

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String Orientation and Level Structures
10 votes

Dylan's nice paper at https://arxiv.org/abs/1507.05116 answers this question.

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Is $\mathbb{H}P^\infty_{(p)}$ an H-space?
9 votes

Sorry for dredging up this question, but here is another argument (at least for $p$ odd, but maybe you don't need this) that came up while thinking about an unrelated problem. If $\mathbf{H}P^\infty_{(...

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Lecture notes by Mahowald and Unell
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9 votes

I scanned the notes (apologies for the delay). Thanks a lot to Peter May for lending me the notes and for letting me scan them! Here's the link: http://www.mit.edu/~sanathd/mahowald-unell-bott.pdf.

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Surveys of Goodwillie Calculus
9 votes

This is 7 years too late, but this survey (to appear in the Handbook of Homotopy Theory) is a really readable survey of Goodwillie calculus: https://arxiv.org/abs/1902.00803.

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Which $\infty$-groupoids correspond to simplicial abelian groups?
8 votes

$\newcommand{\Z}{\mathbf{Z}} \newcommand{\Mod}{\mathrm{Mod}}$Note that if $\Mod^{\geq 0}_\Z$ denotes the category of connective $\mathrm{H}\Z$-module spectra, then by the Dold-Kan correspondence and ...

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Homotopy groups of fiber products
8 votes

Let $X\times_B Y$ denote the homotopy pullback in spaces of maps $X,Y\to B$. (It's not clear whether you mean the ordinary pullback or the homotopy pullback; if $X$ or $Y$ is compact, then $f$ or $g$ (...

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Is the mod-2 Moore spectrum a retract of a shift of its tensor square?
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6 votes

The mod $2$ cohomology of $S^0/2 \wedge S^0/2$ is a $\mathbf{F}_2$-vector space on generators in degrees 0, 1, 1, and 2. The classes in degrees 0 and 2 are connected by a nontrivial $\mathrm{Sq}^2$, ...

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Universal property of $\mathbb S[z]$ and $E_\infty$-ring maps
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6 votes

$\newcommand{\E}{\mathbf{E}}$Dylan answered question 3 (and hence question 1) in the comments, but here's another equivalent way to see it: $\E_\infty$-maps $S^0[z]\to R$ with $R$ a discrete ring (i.e....

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Definition of $E_n$-modules for an $E_n$-algebra
6 votes

$\newcommand{\E}{\mathbf{E}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\cc}{\mathcal{C}}$Here's one way to think about $\E_n$-modules. Let $R$ be an $\E_n$-ring (in a presentable symmetric monoidal ...

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The connective $k$-theory cohomology of Eilenberg-MacLane spectra
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6 votes

Charles Rezk already answered this in the comments; I'll just expand on what he wrote. This paper discusses what's now known as Mahowald-Rezk duality; this is a version of Anderson duality that takes ...

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Spectral and derived deformations of schemes
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6 votes

In general, these are incredibly hard questions. It seems to me that one natural question to ask (if you are interested in $\pi_0$ of ring spectra) would be about understanding even periodic $\mathbf{...

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Any news about equivalences of periodic triangulated or $\infty$-categories?
5 votes

$\newcommand{\QCoh}{\mathrm{QCoh}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\LMod}{\mathrm{LMod}} \newcommand{\spec}{\mathrm{Spec}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\Z}{\mathbf{Z}} \...

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commutative "subalgebras" of associative ring spectra
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5 votes

Let $A$ be an $\mathbf{E}_1$-ring, and let $x\in \pi_n A$. There are two distinct cases to consider. First, if $n = 0$, then the answer to your question is that $x$ is in the image of an $\mathbf{E}_1$...

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A question about maps of spectra
5 votes

If $X$ has homotopy groups in dimensions above $n-1$, then $\mathrm{Map}(X, Y) = \mathrm{Map}(X, \tau_{\geq n} Y)$ for all $Y$. Similarly, if $Y$ has homotopy groups only up through dimension $n$, ...

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Action of fundamental group on homotopy fiber
4 votes

Let $f:E\to B$ be a map of based spaces, and let $F$ be the homotopy fiber. Here is another way of constructing the action of $\Omega B$ on $F$. By definition, there is a homotopy pullback square $$\...

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Relationship between fans and root data
4 votes

Not an answer, but: you can construct a fan from a root system. Let $R$ be a root system in an Euclidean space, and let $\Lambda_R$ be the root lattice with dual lattice $\Lambda_R^\vee$. The fan $\...

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Tannakian criterion for reducedness of Tannakian dual group
4 votes

Yes, and one could argue as follows, at least if the group scheme is assumed to be of finite type (so it is an algebraic group), and the base field is perfect. Recall that if $k$ is a field of ...

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Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum
4 votes

I'm a little confused by your question. You seem to be implying that the Madsen-Tillmann spectra are not Thom spectra, but this is not true: the definition of the spectrum $MTG(n)$ (for $G = O,SO,U$) ...

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How to visualize local complete intersection morphisms?
4 votes

As Dan Petersen said the comments, an lci morphism $f:X\to Y$ is precisely one that factors Zariski-locally as $X\to Y\times \mathbf{A}^n \to Y$, where $X\to Y\times \mathbf{A}^n$ is a regular ...

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If $R$ is an etale extension of $\mathbb Z$, then $R = \mathbb Z^n$?
4 votes

Let $X$ be a normal integral scheme. Then $\pi_1^\mathrm{et}(X)$ is isomorphic to $\mathrm{Gal}(K(X)^\mathrm{un}/K(X))$, where $K(X)^\mathrm{un}$ is the compositum of all finite extensions $F$ of $K(X)...

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Morava $E$-theory source of reading
4 votes

Lurie's notes at http://www.math.harvard.edu/~lurie/252x.html and Rezk's notes http://www.math.uiuc.edu/~rezk/hopkins-miller-thm.pdf explicitly discuss Morava E-theory. Eric Peterson's book project at ...

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