Marc Hoyois
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Is the fundamental group functor a left-adjoint?
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39 votes

The problem is that there are not a lot of actual colimits in the homotopy category of (connected) CW complexes, so knowing that $\pi_1$ preserves them (which is true) is pretty much useless. The ...

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Why is the motivic category defined over the site of smooth schemes only?
25 votes

It's worth noting first that smooth schemes are essentially the smallest possible category from which one can define the motivic homotopy category: to make sense of $\mathbb A^1$-homotopy and of the ...

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When is a topological space the homotopy colimit of an open covering?
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19 votes

It is true in complete generality that $X$ is the homotopy colimit of $C_U$ (and hence that the fat realization computes the homotopy colimit in this case). This is a special case of Lurie's version ...

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Which limits commute with filtered colimits in the category of sets?
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19 votes

Categories $J$ such that limits of shape $J$ commute with filtered colimits in sets are called L-finite. There are several known characterization of them: see the nLab page about it. The page refers ...

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Category of motivic spectra
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18 votes

Here's a direct link to the book by Hovey–Palmieri–Strickland. The category of motivic spectra is known to satisfy the axioms of Definition 1.1.4 in the book when the base is a countable field of ...

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$BG$ the stack, $BG$ the simplicial presheaf
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16 votes

The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player,...

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Can we define homotopy groups using Tannakian categories
16 votes

There certainly is a notion of higher Tannakian category which would have meaningful higher homotopy groups. I'm not sure how much of the theory has been worked out already, but higher Tannakian ...

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Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?
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14 votes

ETA The answer is yes in general. Replace 2 below with a reference to HTT, Prop. 7.1.5.8. Since this has been open for a while, let me give a partial answer which hopefully is already interesting: I ...

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Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?
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14 votes

It doesn't. For instance, the $\infty$-category of spectra is the colimit of the tower $$ \mathcal{S}_* \stackrel\Sigma\to \mathcal{S}_* \stackrel\Sigma\to ... $$ in $Pr^L$, but its colimit in $Cat_\...

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The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?
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14 votes

(1) is true if $char(k)=0$. This follows from a combination of results. First of all, it is true over any field that the spectrum $MGL$ is connective, which means that $$MGL^{p,q}(X)=0$$ if $p>q+...

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Perfect chain complexes
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13 votes

In modern language, one would say that $D_{qcoh}(-)$ is a sheaf of $(\infty,1)$-categories on the scheme $X$ (so "homotopy stack" = "sheaf of $(\infty,1)$-categories"). If $X$ is affine, or more ...

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Realization Functor From $SH$ to Derived Category of $Gal$-Modules
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13 votes

The most general functor of this form was constructed by Ayoub in La réalisation étale et les opérations de Grothendieck. Ayoub considers the ∞-category $DA^{et}(S,\Lambda)$ which is defined exactly ...

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Are there continua in $\infty$-topoi?
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12 votes

Every contractible finite CW complex $X$ satisfies these conditions. This follows from results in Section 7.3 of HTT and Appendix A of HA: we have $Shv(X) \otimes Shv(X)=Shv(X\times X)$ since $X$ is ...

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What's with equivariant homotopy theory over a compact Lie group?
12 votes

Regarding 2, there is no difficulty in defining $G$-spectra in the setting of $\infty$-categories. The only complications I can think of are that (1) the orbit category $\mathrm{Orb}^G$ is now an $\...

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Sheaves of complexes and complexes of sheaves
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12 votes

I'm going to restrict the discussion to Grothendieck abelian categories, because I'm not sure what can be said more generally. The main reference for what follows is Appendix C in Lurie's book ...

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What is the negative cyclic homology of a smooth projective variety?
12 votes

There are conceptually simple definitions, but they require a more symmetric definition of Hochschild homology. The Hochschild homology of $X/k$ (with coefficients in $\mathcal O_X$) is the homology ...

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Diffeology as a sheaf on the site of smooth manifolds
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12 votes

This would be the "comparison lemma" from SGA 4-1, Éxposé III, Thm. 4.1: if $C$ is a full subcategory of a site $D$, equipped with the induced topology, and if every object of $D$ is covered by ...

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Derived version of equivalence between motives and representations of Motivic galois groups?
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11 votes

Let $k$ be a field and $\operatorname{DM}_{gm}(k)_{\mathbb Q}$ the ∞-category of rational geometric motives over $k$. A mixed Weil cohomology theory induces a symmetric monoidal exact functor $$ R: \...

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Is the site of (smooth) manifolds hypercomplete?
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11 votes

I think your idea to reduce the question to small slice topoi works perfectly. I will use it to show that every sheaf on $Man$ (either the continuous or the smooth version) is the limit of its ...

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A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (necessarily) algebraically closed field
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11 votes

The only reference I know for Künneth theorems in this generality are SGA4 and SGA4.5. Specifically, SGA4 has theorems for étale cohomology with proper support (which hold in ridiculous generality), ...

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Tensor products of $\mathbb{E}_\infty$-spaces
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10 votes

The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful ...

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$2$-fiber product is a scheme then map of stacks is representable
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10 votes

This is not true even if $\mathcal X$ is an Artin stack. For example, let $G$ be a smooth group scheme over the base $T$, and let $\mathbf BG$ be its classifying stack (the category of $G$-torsors ...

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Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
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10 votes

This is true, yes. More generally, if $X$ is a scheme and $F$ is a locally constant étale sheaf of finite abelian groups on $X$, then $$ H^1_{et}(X,F) = H^1(\Pi_1^{et}(X), \tilde F), $$ where $\Pi_1^{...

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When is the pullback in Chow groups defined?
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10 votes

You have to distinguish between pullbacks of cycles, pullbacks on Chow groups, and pullbacks of relative cycles. You cannot always pull back cycles. If $f: Y\to X$ is a morphism of (arbitrary) ...

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Homotopy left-exactness of a left derived functor
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10 votes

I do not know the answer for a general Quillen adjunction, but I will attempt to give a complete answer in the case you're interested in, when the adjunction $(F,G)$ is of the form $(f_!,f^\ast)$ for $...

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FIltered colimits of truncated objects in $\infty$-topoi
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10 votes

I believe the answer is YES and, more generally, that $\tau_{\leq n}\mathcal{C}\subset\mathcal{C}$ preserves filtered colimits for any $\infty$-topos $\mathcal{C}$. For the $\infty$-topos of $\infty$-...

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Example of a locally presentable locally cartesian closed category which is not a topos?
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9 votes

Every Grothendieck quasitopos is presentable and locally cartesian closed. These are categories of separated presheaves on a site. The simplest example of a site whose separated presheaves do not form ...

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Is $MGL$ an $H\mathbb{Z}$-algebra?
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9 votes

$MGL$ does not admit a structure of $H\mathbb Z$-module. There are many ways to prove this. As Sean said in the comments, if it were true over $\mathbb C$, topological realization would imply that $MU$...

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motivic t-structure and realisations
9 votes

The image of the inclusion $CHM_k\hookrightarrow DM_k$ is indeed not contained in the heart $M_k$ of the motivic t-structure. $CHM_k$ does map to the heart, but via a different functor, namely, the ...

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Determinantal identities for perfect complexes
8 votes

The formula also holds for perfect complexes. This can be deduced from the case of vector bundles, although it requires a lot of structure in that case. Namely, we need to use the fact that the ...

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