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Let's consider these in characteristic zero, so that we can use differential graded algebras and so that we can model $E_\infty$ things by strictly commutative things and $E_1$ things by associative t …

In general the issue is that the natural map $A \to \tau_{\leq 0} A$ often cannot be compatible with a ring structure, because on the level of homotopy groups or homology groups it acts as a quotient …

There are actually quite a number of other examples. In particular, this is necessary and sufficient for $R$ to be a so-called smashing localization of the sphere $\Bbb S$, and there are several promi …

By definition, the space $GL_1(R)$ is the subspace of $\Omega^\infty R$ consisting of those elements whose path component $\alpha \in \pi_0(\Omega^\infty R) = \pi_0(R)$ is a unit in $\pi_0(R)$. If $\ …

Your description of the functor is correct. The suspension spectrum functor is homotopy colimit preserving, as you say. Smashing with A is left adjoint to the forgetful functor from A-modules to spect …

Limit-preservation is the content of section 3.2.2 of Higher Algebra (see particularly Corollary 3.2.2.5). Preservation of sifted colimits is in section 3.2.3 (see particularly Corollary 3.2.3.2). As …

Yes, this is true. This kind of property for a model category is a consequence of what is sometimes abusively called the "SM7" axiom for the enrichment: In $R$-modules, suppose that $A \to B$ is a …

There are no left-unital multiplications on E. If there were, then for any element $x$ in $\pi_n(E)$, we would have $x+x = 1 \cdot x + 1 \cdot x = (1+1) \cdot x = 0$ because all elements in $ \pi_0 E$ …

No, this is not true, and nothing like this is expected. The exact sequence $\Bbb Z/2 \to \Bbb Z/4 \to \Bbb Z/2$ gives rise to a fiber sequence $H\Bbb Z/2 \to H\Bbb Z/4 \to H\Bbb Z/2$ of Eilenberg--Ma …

This is an elaboration on Lennart's comment. This can be made to come from a Quillen equivalence. Here are the ingredients you'd usually need to show it. (Sorry, I don't have my copy of EKMM handy …

Here is a little context to maybe complement Tom's and Nick's answers. The definition (2) in terms of being flat over $\pi_* \Bbb S$ is new - it's a specialization of a definition of flatness over $R …

Here is something that I think is reasonably difficult to get around. As you observe, the unit is the initial object of commutative monoids, and so your request includes that the unit is cofibrant. Su …

Suppose $A$ is any nontrivial connective ring spectrum, with right module $P = A \oplus A[1]$, and let $B = End_A(P)$. The ring $B$ satisfies $$B \simeq A \oplus A[1] \oplus A[-1] \oplus A$$ and in pa …