33 votes
Accepted

What is the relationship between connective and nonconnective derived algebraic geometry?

Here is an example of a nonconnective $E_\infty$ ring spectrum which, I think, illustrates a key problem. (A more extensive discussion of this phenomenon occurs in Lurie's DAG VIII and in a paper by ...
Tyler Lawson's user avatar
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33 votes
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What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

Lurie characterizes the symmetric monoidal structure on $\mathsf{Sp}$ by a universal property (HA.4.8.2.19): it is uniquely determined up to a contractible space of choices by the property that $S^0$ ...
Dylan Wilson's user avatar
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29 votes
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Derived categories and $\infty$-categories necessary for condensed mathematics

There are several questions (implicit) here. In the texts as they are written, how much knowledge on derived categories (as triangulated categories, or as stable $\infty$-categories) is assumed? ...
Peter Scholze's user avatar
22 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

The question used the phrase "still needed." This is a very loaded term, and the answer that you give will depend very strongly on how you interpret it. If, as Dylan does, we interpret this as asking ...
20 votes
Accepted

Is the $\infty$-category of spectra “convenient”?

My colleague Dylan answered first (I keep telling him not to spend too much time on this toy :) but I both agree and disagree with his "Yes of course". The same words are used with different meanings ...
Peter May's user avatar
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20 votes
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Describing fiber products in stable $\infty$-categories

In fact what you need is that your ∞-category is additive (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories ...
Denis Nardin's user avatar
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19 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

At the risk of starting some kind of (un?)civil war, let me expand on my comments. First and foremost, let's address the interpretation of the question. The OP asks "do we need a model category of ...
18 votes
Accepted

Where to find the correct result in Higher Algebra, incorrect reference

The correct reference is 6.1.4.14. (And the hypothesis of 6.1.6.27 should refer to countable limits and colimits, rather than finite limits and colimits.)
Jacob Lurie's user avatar
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17 votes
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What is the free symmetric monoidal $\infty$-category on one object?

Yes, it is the same as $\mathbb{F}$. As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints ...
Chris Schommer-Pries's user avatar
16 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

The original question has been answered in the sense that there are people who are confident to prove every statement about spectra they care about without recourse to models or model categories of ...
14 votes

What is the relationship between connective and nonconnective derived algebraic geometry?

As Tyler pointed out, it is "too easy" to be representable in the non-connective world. This might sound good, but it comes at the cost of geometric intuition. It is related to the fact that ...
AAK's user avatar
  • 5,841
14 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

Given that Sp is better behaved than all other existing models of spectra No, Sp is not better behaved than other models. The reason that it seems to be is because all operations in Sp (e.g., Ω^∞, Σ^∞...
13 votes
Accepted

The universal property of the unseparated derived category

Yes, both of these statements are true (I thought they were in the book, but I can't seem to find them now). Here is a proof sketch. Let's start with the case described in 2). Let $\mathcal{C}$ be ...
Jacob Lurie's user avatar
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13 votes
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Does formation of the derived $\infty$-category preserve pushouts?

A hands-on explanation: Relative tensor products like $B\otimes_AC$ are computed as the colimit of the simplicial object $B\otimes A^{\otimes \bullet} \otimes C$. The functor $\mathsf{Mod}_{(-)}: \...
Dylan Wilson's user avatar
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13 votes
Accepted

Is the forgetful functor $\mathrm{Mod}_R \mathrm{Sp} \rightarrow \mathrm{Sp}$ faithful?

$U_R$ obviously preserves delooping, so if that were the case, because $\pi_0 map(X,Y) = \pi_1 map(X, \Sigma Y)$, you would also get an isomorphism on $\pi_0$, so an equivalence of mapping spaces. In ...
Maxime Ramzi's user avatar
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13 votes

What is the dual of the stable infinity category of perfect complex on smooth proper variety?

It is self-dual. In general the dual of a smooth proper $R$-linear $\infty$-category $C$ is always $C^{\operatorname{op}}$, but for a scheme $X$ we have $\operatorname{Perf}(X) = \operatorname{Perf}(X)...
Marc Hoyois's user avatar
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12 votes
Accepted

What is the homotopy category of the sphere spectrum?

This is the groupoid given by the 1-truncation $\tau_{\leq 1}(QS^0)$. This groupoid has $\mathbb Z$-many objects (since $\pi_0^s = \mathbb Z$), and each one has automorphism group $C_2$ (since $\pi_1^...
Tim Campion's user avatar
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11 votes

What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

Let me add a short observation to Dylan's fantastic answer. There is indeed a more concrete construction of the symmetric monoidal structure on the $\infty$-category of spectra: it is the localized ...
Denis Nardin's user avatar
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11 votes
Accepted

Proper model category of simplicial rings revisited

This paper proves some things about left properness for categories of simplicial algebras. The context of the paper is in terms of "algebras for a simplicial algebraic theory", which certainly ...
Charles Rezk's user avatar
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11 votes
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Definition of $E_n$-modules for an $E_n$-algebra

$E_n$ algebras have compatible multiplications for every way of placing a bunch of elements into a collection of balls in $\mathbb{R}^n$. A module for an $E_n$ algebra has an action for every way of ...
Noah Snyder's user avatar
  • 27.8k
11 votes

Is the $\infty$-category $N_{dg}(\mathrm{Ch}(\mathcal{A}))$ presentable?

This fails already with the category of abelian groups. If the dg-nerve of the dg-category of chain complexes of abelian groups were presentable, then the associated triangulated category would be ...
D.-C. Cisinski's user avatar
11 votes
Accepted

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

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 ...
Marc Hoyois's user avatar
  • 8,672
11 votes

Why the Bousfield localization of spectra at topological K group is important?

(I'm very surprised that the following hasn't been mentioned in the comments so far -- I thought it was conventional wisdom!) The main reason to study $K(1)$-local homotopy theory is that it is the ...
Tim Campion's user avatar
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10 votes
Accepted

understanding the definition of $\infty$-operad of module objects

I know this question is a little old but I just came across it. Roughly, as you say, from this data you get an object $v$ of $O^\otimes$ and an algebra $A$ of $Alg_{/O}(C)$. However, you also get an ...
Tyler Lawson's user avatar
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10 votes
Accepted

Grading ring spectra over the sphere spectrum

One of the default examples of ordinary graded commutative rings is the polynomial ring $\mathbf Z[t]$. Let us first examine the analogue of that, and then see where else that leads! 1. $S$-grading on ...
A Rock and a Hard Place's user avatar
9 votes
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Dual objects in the $\infty$-category of spectra

As requested, the comments turned into answers: The dualizable objects in spectra are precisely the finite spectra (i.e. spectra of the form $\Sigma^{-k}\Sigma^{\infty}X$ where $X$ is a finite ...
9 votes
Accepted

Is the $E_\infty$-structure on the cochain complex of a $K(G,n)$ readily understandable?

There is a very general framework that gives the $E_\infty$ structure on cochains in McClure and Smith's "Multivariable cochains and little $n$-cubes", with formulas comparable to the Alexander-...
Tyler Lawson's user avatar
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9 votes
Accepted

Is the de Rham complex in characteristic $p$ a CDGA?

For a smooth and proper scheme $X$ over a field $k$ of characteristic $p$, the $E_\infty$-algebra $R\Gamma(X, DR_{X/k})$ over $k$ is not represented by a $k$-cdga. Indeed, if it were, then most of the ...
Meric's user avatar
  • 106
9 votes
Accepted

How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?

By Corollary HA.4.8.5.20, the functor from $\mathbb{E}_{n+1}$-algebras to $\mathbb{E}_n$-monoidal categories and colimit-preserving, $\mathbb{E}_n$-monoidal functors is fully faithful. (Notice that ...
Dylan Wilson's user avatar
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