All Questions
Tagged with derived-algebraic-geometry homotopy-theory
21 questions with no upvoted or accepted answers
27
votes
0
answers
1k
views
Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 64k
11
votes
0
answers
887
views
How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $\mathbb{E}_\infty$-rings)?
Jacob Lurie has extensively developed derived algebraic geometry in the setting of $\mathbb{E}_\infty$-ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in ...
- 11.8k
9
votes
0
answers
288
views
Every Spectral Deligne-Mumford stack satsifies fpqc descent?
In SAG Remark 6.3.3.8, Lurie asserts that if we have a representable (by Spectral Deligne-Mumford stacks) natural transformation $X\to Y$ where $Y$ is a functor satisfying fpqc descent, then so too ...
- 19.6k
7
votes
0
answers
161
views
Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?
Is it always possible to write a derived manifold as a homotopy colimit of principal derived manifolds (i.e. zero sets of smooth functions)? This is true for schemes and derived schemes, so it seems ...
- 31
6
votes
0
answers
170
views
New investigations on Homotopical Algebraic Contexts
Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II.
These are general abstract ...
- 177
6
votes
0
answers
226
views
Applications of spectral Artin representability?
The spectral Artin representability theorem says that a functor $X:CAlg^{cn}\rightarrow S$ from the $\infty$-category of connective $E_{\infty}$-rings to the $\infty$-category of small topological ...
user74900
5
votes
0
answers
395
views
Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
- 889
5
votes
0
answers
447
views
Infinity categories with an action of a simplicial group
Recent papers in derived algebraic geometry use a notion of $S^1$-actions on infinity categories. I think I understand what this "should" be and how to calculate with it; however, I can't find a much ...
- 1,423
5
votes
0
answers
225
views
Does the Amitsur complex have a universal property?
The question is essentially the title. In other words, is there some universal property that the Amitsur complex for a morphism of rings $\phi:A\to B$ satisfies as a cosimplicial ring, or cosimplicial ...
- 10.5k
4
votes
0
answers
153
views
Preorientation of additive formal group
In "A Survey of Elliptic Cohomology", Section 3.2, Lurie asserts that the preorientations of the additive formal group $\widehat{\mathbf G}_a$ over $\mathbf Z$ are classified by the $\mathbb ...
- 2,011
4
votes
0
answers
118
views
Does the ∞-category of Derived/Spectral schemes admit all colimits over constant diagrams?
In the case of ordinary schemes, all coproducts exist, so given any constant diagram $D_S:C\to \operatorname{Sch}$, the colimit over $D_S$ is isomorphic to the coproduct of $S$ over the connected ...
- 493
4
votes
0
answers
477
views
DAG applied to homotopy theory: how to reach research level?
It is my dream to do research on applications of spectral algebraic geometry in homotopy theory one day. Specifically, giving a more uniform treatment for the results proved via scary computations (of ...
3
votes
0
answers
225
views
Derived $\infty$-category of quasi-coherent sheaves on schemes
Let $X$ be a scheme. On the one hand, we have the derived $\infty$-category constructed from the abelian category of quasi-coherent sheaves on $X$. On the other hand, we can define the stable $\infty$-...
- 151
2
votes
0
answers
116
views
Quasicompact quasiaffine classical schemes are nonconnectively-affine
In this answer to What is the relationship between connective and nonconnective derived algebraic geometry? I learned that any quasicompact open subscheme of an affine scheme is affine in the sense of ...
- 448
2
votes
0
answers
354
views
Higher-order HKR theorems?
Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...
- 64k
2
votes
0
answers
189
views
Is the homotopy limit of derived schemes along affine maps a derived scheme?
The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes.
I'd ...
- 301
2
votes
0
answers
104
views
Linearity of a dg category $C$ over $HH^0(C)$
Let $C$ be a pre-triangulated dg-category over a field $k$ whose Hochschild cohomology groups $\operatorname{HH}^*(C)$ are concentrated in non-negative degree (cohomologically). Is $C$ Morita ...
- 3,758
2
votes
0
answers
268
views
Interesting examples of large, accessible, non-presentable $\infty$-categories?
What are some interesting examples of accessible $\infty$-categories
which are not presentable and not small?
By interesting I mean a category which comes up naturally in a certain context and in a ...
- 7,799
2
votes
0
answers
277
views
classifying space of algebraic groups
Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a Borel pair $(B,T)$.
Let $BG$ be the classifying space of $G$.
Can we say that $BG$ is the homotopy colimit of all $BP$ for $P$ a ...
- 3,472
2
votes
0
answers
166
views
How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?
First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of $...
- 9,019
1
vote
0
answers
281
views
Étale homotopy type of (derived) loop space
A feature of derived algebraic geometry is that we have internal homs. Furthermore, we can think of $B\mathbb{Z}$ as the derived algebraic geometric analogue of $S^1$. Thus we have an analogue of the ...
- 4,418