# Tag Info

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 ...
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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 $S^0$ ...
• 12.8k

### 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 ...
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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 ...
• 15.8k
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### 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
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• 12.8k
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### 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 ...
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• 51.1k

### 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
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### 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 ...
• 17k
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### 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 ...
• 17k

### 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
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### 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 $Ab(Sh(X))$ ...
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### $k$-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 ...
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### 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
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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 ...
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### 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 ...
• 12.2k
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### 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 ...
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### 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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