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Nov
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comment Realization Functor From $SH$ to Derived Category of $Gal$-Modules
@MikhailBondarko If $S$ is excellent and $p$ is invertible on $S$, then for every strictly henselian local ring $A$ of $S$ and every $x\in Spec(A)$, $cd_p(\kappa(x))=dim(\overline{\{x\}})<\infty$. This finiteness condition, for all $p$ dividing $n$, is the assumption of Ayoub's rigidity theorem.
Nov
21
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Nov
21
comment Realization Functor From $SH$ to Derived Category of $Gal$-Modules
I should add, it's only a technical matter to make $p\mapsto p_!p^!\Lambda$ into an (op?)lax symmetric monoidal functor, and it is known to be strict symmetric monoidal when $S$ is excellent, by work of Gabber. When $S$ is a field, however, this was proved by Deligne in SGA4½.
Nov
21
answered Realization Functor From $SH$ to Derived Category of $Gal$-Modules
Oct
21
comment stackifickation of BG
This is probably somewhere in Principal ∞-bundles by Nikolaus, Schreiber, and Stevenson. Classical references either use hypercover stackification or fail to note that stackification preserves $n$-truncated presheaves, so they're not much help here.
Oct
19
comment stackifickation of BG
The stackification sends $N$ to the nerve of the groupoid of principal $G$-bundles over $N$ equipped with a $G$-equivariant map to $M$. It's clear that this is a stack, and I think it's what you get from $X$ by applying the plus construction just once.
Sep
26
comment Smash product of spheres in $\mathbf{SH}$ and product in cohomology
@Tintin This follows from the universal property of $SH$, a reference for which is Higher Algebra, Cor. 1.4.4.6. More concretely, if you take $T$-$S^1$-prespectra in simplicial presheaves as a model for $SH(X)$, then the functor sending a simplicial spectrum to $\Sigma^\infty_T$ of the corresponding constant presheaf is left Quillen for appropriate model structures.
Sep
25
comment Smash product of spheres in $\mathbf{SH}$ and product in cohomology
@Tintin Being the homotopy category of a stable model category, $SH(X)$ is tensored over $SH$. The functor $SH\to SH(X)$ I'm talking about is $E\mapsto E\otimes 1_X$. You may define spheres so that the horizontal maps are equalities, why is that a problem?
Sep
25
comment Smash product of spheres in $\mathbf{SH}$ and product in cohomology
The sign should be $(-1)^{ab}$. This square is a functorial image of the analogous square in the ordinary stable homotopy category, where the commutativity boils down to computing the degree of reflections on spheres.
Sep
25
comment Smash product of spheres in $\mathbf{SH}$ and product in cohomology
The diagram for $a=b=0$ does commute by the axioms for a symmetric monoidal category.
Sep
5
comment $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras
Barr-Beck doesn't require creation, existence+preservation suffices. Your guess as to what $\phi$ is is as good as mine.
Sep
4
comment About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
@user40276 I don't see how you can conclude from this that Sch ⊂ LRS preserves all limits. What fails for LRS is that the category of sheaves of rings with local stalks over a fixed space is not cocomplete.
Sep
4
comment $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras
I believe 2 and 3 are answered in DAG5, Prop 4.1.11, modulo the Barr-Beck theorem from Higher Algebra (there is no need for $R$ to be a field, it seems).
Sep
2
revised About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
added 124 characters in body
Sep
1
comment About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
@user40276 I should have said that LRS ⊂ RS creates colimits. This is not the statement of Prop 1.6 but that's how it's proved. For limits it's clear for instance that $Spec(k)\times Spec(k')=\emptyset$ in LRS if $k$ and $k'$ are fields with different characteristics. I did mean section 5.1 for the statement that Sch ⊂ LRS preserves finite limits, which is also proved in the paper you linked to, but I don't think it's as formal as you suggest. It's definitely not true that any map from an affine scheme factors through an affine open...
Aug
31
comment About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
In your last example, a pullback of fields will be a field... In fact LRS ⊂ RS preserves colimits, though of course not limits. This is Prop 1.6 in "Groupes algébriques" by Demazure and Gabriel. On the other hand, Sch ⊂ LRS preserves finite limits (section 5.1 in loc. cit.). I think it also preserves cofiltered limits with affine transition morphisms.
Aug
28
comment A “universally non Hypercomplete” $\infty$-topos?
As you say, any $1$-excisive functor $\mathcal{S}^\mathrm{fin}_* \to \mathcal{S}$ belongs to this localization, since $(*,S^0)$ is ∞-connected, but I don't know if there's an unpointed analog of this (what's $\mathrm{Exc}^1(\mathcal{S}^\mathrm{fin},\mathcal{S})$ anyway? torsors under bundles of spectra?). In any case, I don't see why the reverse inclusion should hold, but I'm no expert in calculus either...
Aug
28
comment A “universally non Hypercomplete” $\infty$-topos?
Right, the classifying ∞-topos for (pointed) $n$-connected objects is the left exact localization of $\mathrm{Fun}(\mathcal{S}^\mathrm{fin}_{(*)}, \mathcal{S})$ generated by the map $\tau_{\leq n} \to *$, and for $n=\infty$ it's their intersection.