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Yes this is true. This follows implicitly from the fact that the $E_1$-structure on $Mf$ can be identified with the canonical $E_1$-structure on $\mathrm{colim}_Xf$ (see for example here) and that is …

Recall that by representability of cohomology plus the Yoneda lemma, a cohomology operation $H^i→H^j$ is the same thing as a map $$ K(\mathbb{F}_p,i)→K(\mathbb{F}_p,j)\,.$$ Moreover, the suspension is …

Ravenel in his Complex Cobordism and Stable Homotopy Groups of Spheres attributes this result to Stong, in Determination of $H^*(BO(k,⋯,∞),Z_2)$ and $H^∗(BU(k,⋯,∞),Z_2)$, but looking at that paper (wh …

What you're missing is that $[\mathbb{S},\mathbb{S}]=\mathbb{Z}$. Let now $R$ be the infinite matrix of integers representing $\rho$. Note that since $\rho$ takes value in $\bigoplus_{I_1}\mathbb{Z}\s …

This is true. To prove it I will use the fact that maps of parametrized spectra can be computed as natural transformations of functors from $X$ into the $\infty$-category of spectra (cfr. this paper) …

NO such a map does not exist. Thanks to Eric Peterson for making me realize that the argument carries through even if the map is not a map of algebras. By rigidity,you can only consider the case whe …

In the case where $\mathcal{C}$ is presentable, this is constructed in proposition 2.9 of Robalo, Marco, $K$-theory and the bridge from motives to noncommutative motives, Adv. Math. 269, 399-550 …

A simple way of seeing it is to explicitly spell out what we mean when we say that a spectrum is "presented" by a prespectrum. To say that that a spectrum $E$ is presented by $$(E_0,E_1,...)$$ means t …

This is always true, even without the hypothesis of stability. In an ∞-category a fiber sequence $X\to Y\to Z$ is a pullback square $$\require{AMScd} \begin{CD} X @>>> Y\\ @VVV @VVV \\ * @>>> Z \end{C …

I'm not sure I can answer to the general question, but I can explain why the Moore spectrum $\mathbb{S}/p$ shows up in the discussion of Bousfield localizations. This is just the cofiber of multiplic …

The key here is that $SH^{eff}(S)$ is closed under suspensions, so there's an inclusion $j_{n+1}:\Sigma^{n+1}_T SH^{eff}(S)\subseteq \Sigma^n_T SH^{eff}(S)$. Hence you can write $i_{n+1}=i_n \circ j_{ …

For the equation you are asking about, you don't want $MA$ to be dualizable (which is lucky, because it's not), you want $MA\wedge X_+$ to be dualizable as an $MA$-module. This is true in the situatio …

$S^1$-spectra Let me first show it when the category is $SH^{S^1}(k)$, that is the (∞-)category of $\mathbb{A}^1$-invariant sheaves of spectra. Then the heart of the t-structure is precisely the categ …

This is worked out in part 2 of Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). …

Let me first write what happens for classical cobordism. You are basically asking whether the map $\operatorname{MU}\to H\mathbb{Z}$ factors through the projection $\operatorname{ku}\to H\mathbb{Z}$. …