All Questions
18
questions
13
votes
2
answers
520
views
How many automorphisms are there of the category of filtered spectra?
Dold-Kan type theorems tell us that lots of categories are Morita-equivalent to the simplex category $\Delta$. In other words, there are a lot of stable $\infty$-categories which are secretly ...
- 62.6k
6
votes
1
answer
176
views
When is an $\infty$-categorical localization of an additive 1-category enriched in topological abelian groups?
Let $\mathcal A$ be an additive 1-category, equipped with some class of weak equivalences $\mathcal W$. Let $\mathcal A[\mathcal W^{-1}]$ be the localization of $\mathcal A$ at $\mathcal W$ (so $\...
- 62.6k
6
votes
0
answers
233
views
Flatness of objects in a prestable $\infty$-category
I wonder what is the correct concept of flatness of objects in a prestable $\infty$-category with appropriate conditions?
The typical example is the following. Let $R$ be a connective $\mathbb E_1$-...
- 2,446
3
votes
2
answers
301
views
How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2 X$ without splitting an idempotent?
Let $Z$ be an object in a stable (or triangulated/whatever) category $\mathcal C$. I believe it follows from Thomason's theorem (see The classification of triangulated subcategories) that the ...
- 62.6k
4
votes
0
answers
177
views
Spanier-Whitehead dual of space of natural transformations
Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).
...
14
votes
2
answers
700
views
When is a stable $\infty$-category the stabilization of an $\infty$-topos?
Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ ...
- 62.6k
8
votes
1
answer
467
views
Is there a Dold-Kan theorem for circle actions?
There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 ...
- 62.6k
4
votes
0
answers
124
views
What is the colimit closure of the finite endomorphism spectra?
$\newcommand{\colim}{\operatorname{colim}}\newcommand{\finend}{\operatorname{finend}}$Let $F$ be a finite spectrum. Then $\operatorname{End}(F) = D(F) \wedge F$ is also finite.
Question: Which spectra ...
- 62.6k
4
votes
1
answer
253
views
When is a thick subcategory the preimage of a weak Serre class under a homological functor?
Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $...
- 62.6k
5
votes
0
answers
118
views
Variations on Thomason's equivalence between connective spectra and symmetric monoidal categories
There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces):
Symmetric monoidal categories model all connective ...
- 11.5k
5
votes
1
answer
474
views
Categorical models for truncations of the sphere spectrum
Picard $n$-groupoids are expected to model stable homotopy $n$-types. So far this has been proved for $n=1$ in
Niles Johnson, Angélica M. Osorno, Modeling stable one-types. Theory Appl. Categ. 26 (...
- 11.5k
4
votes
0
answers
195
views
Direct image and infinite suspension
I have a basic doubt regarding infinite suspension functor and the direct image. I write it for schemes but I guess it works the same for the topological setting so I welcome answers also from the ...
- 2,811
11
votes
1
answer
502
views
Generalized "Homology Whitehead" -- How much does stabilization remember?
Classically, the (non-local-coefficients) homology Whitehead theorem says that if $X \xrightarrow f Y$ is a map of simple spaces, and if the induced map $H_\ast(X;\mathbb Z) \to H_\ast(Y;\mathbb Z)$ ...
- 62.6k
7
votes
0
answers
340
views
Example of a tensor triangulated category with two different monoidal t-structures?
What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures?
While I'm at it: is there an example of a tensor ...
- 62.6k
23
votes
5
answers
3k
views
What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?
Alternate formulation of the question (I think): What's a precise version of the statement: "In a stable $\infty$-category, finite limits and finite colimits coincide"?
Recall that a stable $\infty$-...
- 62.6k