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 ...

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. $\...

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

$\newcommand{\H}{\mathrm{H}} \newcommand{\Z}{\mathbf{Z}}$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_\...

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 ...

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 ...

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 ...

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

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 $M$ ...

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_{(...

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.

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.

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

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 $g$ (...

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 $\mathrm{Sq}^2$, ...

$\newcommand{\E}{\mathbf{E}}$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....

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

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 ...

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{...

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

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 $\mathbf{E}_1$...

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 $n$, ...

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 $$\...

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 $\...

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 ...

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 $G = O,SO,U$) ...

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 ...

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)...

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 ...