32 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 ...
  • 48.1k
31 votes
Accepted

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 $S^0$ ...
  • 12.8k
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

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 ...
  • 15.8k
19 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 ...
  • 29.2k
18 votes
Accepted

Can we construct a Baas-Sullivan presentation of TMF?

There are a couple of possible variant questions of this, and I'm not quite sure which is appropriate. The first question question is whether, without knowledge of the functor $T$, we could construct $...
  • 48.1k
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.)
17 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 ...
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 ...
15 votes
Accepted

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 ...
14 votes
Accepted

Obstructions for $E_n$-algebras

Let me expand a little on what Qiaochu and Craig mentioned. If you want an obstruction theory for building an uber-gadget, you'll need (i) an algebraic approximation to such gadgets, and (ii) a way ...
  • 12.8k
13 votes

What are $( \infty , n)$-categories useful for?

In my humble opinion, the fact that such a structure appears naturally on bordisms between manifolds (eventually with some additional geometric structure like framings) is already a very good ...
  • 1,559
13 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

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}_{(-)}: \...
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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 ...
  • 9,690
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)...
  • 7,974
12 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 ...
  • 5,609
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^...
  • 51.1k
11 votes

Is there an analog of the Barratt-Eccles construction for group-like E_∞-spaces and E_∞-ring spaces?

There are serious problems making your ideas coherent here! The notion of operad was in large part intended to model kinds of algebras whose laws do not involve repeated variables, do not involve ...
  • 29.2k
11 votes
Accepted

Associative Ring Spectra and Derived Completion

If by ``effective monomorphism'' you mean the categorical dual of the condition of being an effective epimorphism, then it is not (or at least not obviously) equivalent to the statement that A can be ...
11 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 ...
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 ...
  • 15.8k
10 votes
Accepted

unbounded derived category of a $\infty$-topos

Note: answer corrected thanks to the comments of Dylan Wilson and Marc Hoyois. Let $X$ be a topological space. Its derived category $D(X)$ is the derived category of the abelian category $Ab(Sh(X))$ ...
10 votes
Accepted

$k$-Disk algebras versus $E_k$ algebras

There is an unfortunate clash of terminologies here. Traditionally, the little discs operad comes in two variants: the "usual" $\mathtt{D}_n$: the space of arity $r$ operations consists of embeddings ...
10 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 ...
  • 25.9k
10 votes
Accepted

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 ...
  • 26.7k
10 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 ...
10 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 ...
  • 7,974
10 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 ...
  • 51.1k

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