All Questions
Tagged with derived-categories homotopy-theory
28 questions
3
votes
0
answers
97
views
Examples of tensor-triangulated categories not satisfying the local-to-global principle
From now on, we will consider only rigid-compactly generated tensor-triangulated categories. Let $(\mathcal{T}, \otimes, 1)$ be one of these categories, it is known that the thick tensor ideals of ...
- 767
1
vote
1
answer
346
views
Homotopy pullback is right adjoint in the derived category
Let $f: X \to Y$ be a map of CW-complexes with continuous maps as morphisms.
How would one show that homotopy pullback $\mathcal D/Y → \mathcal D/X$ is right adjoint?
Here $\mathcal D$ is the derived ...
user30211
2
votes
0
answers
311
views
Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy
$\newcommand\Ab{\mathrm{Ab}}\newcommand\ab{\mathrm{ab}}\DeclareMathOperator\Ch{Ch}\DeclareMathOperator\Kom{Kom}\newcommand\ho{\mathrm{ho}}$Let $\Ab$ be the category of finitely generated abelian ...
- 5,230
10
votes
1
answer
2k
views
Computations in condensed mathematics, page 32-34
I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions:
At the last line of pg 32 - it seems to imply that ...
- 661
1
vote
2
answers
726
views
On the link between homology and homotopy
In the last semester I learned homological algebra and higher category theory/homotopy theory.
But I am kind of confused when I try to really understand the link between the two subjects (this is ...
- 546
5
votes
1
answer
564
views
Computation on homotopy colimit cocomplete triangulated categories
I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories.
Question I:The first one concerns a comment by Peter Arndt in this discussion about derived ...
- 5,966
3
votes
0
answers
129
views
Constructing functorial homotopies in derived infinity-category
I'm interested in the following problem : let $\mathcal{C}$ be an $\infty$-category and $\mathcal{D}:=D_\infty(\mathbb{Z})$ the derived $\infty$-category of abelian groups. Consider functors $A, B, C ,...
- 617
7
votes
0
answers
320
views
Derived symmetric powers and determinants
Given a vector bundle $V$ (on a scheme $X$, say), I can form $Sym(V[1])$, the symmetric algebra (in the derived/graded sense) on the shift of $V$; in other words this is the Koszul complex of the zero ...
- 1,961
11
votes
1
answer
2k
views
Derived base change in étale cohomology
Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
- 28.8k
10
votes
1
answer
883
views
$\infty$-categorical understanding of Bridgeland stability?
On triangulated categories we have a notion of Bridgeland stability conditions.
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
- 103
9
votes
0
answers
328
views
Dualizable objects in homotopy category of chain complexes
The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that:
Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...
user95283
3
votes
1
answer
486
views
Spectral sequence for tensor product of complexes
Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X_{\rm ét}$.
I want to compute the hypercohomology:
$$\mathbb{H}^*(X_{\rm ét}, K^{\bullet}\otimes^L_{\...
user113452
17
votes
2
answers
1k
views
Constructive homological algebra in HoTT
I'm curious how much of homological algebra carries over to a constructive setting, like say HoTT (or some other variety of intensional type theory) without AC or excluded middle. There doesn't seem ...
- 6,025
36
votes
3
answers
6k
views
What is a triangle?
So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...
- 10.7k
4
votes
0
answers
76
views
In which sense is the relativization functor "preferred"?
In A characterization of simplicial localization functors and a discussion of DK equivalences Barwick and Kan state that, while there is no preferred localization functor from relative categories to ...
- 651
5
votes
1
answer
552
views
Defining hom spaces in the derived category as limits of hom spaces in the homotopy category
Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof):
A. The following isomorphisms hold:
$$\...
- 7,799
3
votes
1
answer
378
views
Reference for comparison of heart cohomology with standard cohomology
I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations).
Let A,B be two hearts of ...
- 1,280
51
votes
5
answers
5k
views
What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...
- 40.3k
5
votes
1
answer
622
views
Is the injective structure on unbounded chain complexes simplicial?
In Mark Hovey's article Model category structures on chain complexes of sheaves (arXiv:math/9909024) a model structure on the category $Ch(A)$ of unbounded chain complexes for a Grothendieck abelian ...
- 105
7
votes
1
answer
467
views
Are constructible derived categories invariant up to weak homotopy equivalence?
Let $X$ and $Y$ be two topological spaces and $R$ be commutative ring. Let $D_c^b(X, R)$ and $D_c^b(Y,R)$ be their respective bounded derived categories of constructible sheaves of $R$-modules. I ...
- 121
0
votes
0
answers
139
views
D(R) versus Ho(HR)?
Given an algebraic ring, how is its derived category related to the homotopy category of HR modules?
Thanks. This is essentially a reference request, since I know there may be a lot (or nothing) to ...
- 10.5k
2
votes
1
answer
1k
views
derived functors and triangulated categories
If you derive a right exact functor $F$ you get a functor normally denoted by $RF$ on the derived category. Similarly, if you start with a left exact functor $G$ you get a functor normally denoted by $...
- 2,842
3
votes
0
answers
209
views
Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor
Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
- 607
15
votes
4
answers
2k
views
Intuition about the triangulation of a homotopy category K(A)
Let $\cal{A}$ be an additive category. Given a morphism of (cochain) complexes $f:X\rightarrow Y$ we can form the mapping cone $C_f$, which is the complex $X[1]\oplus Y$ with differential given by
$$\...
- 2,814
11
votes
3
answers
1k
views
Compact generation for modular representations
Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by ...
- 24.1k
36
votes
6
answers
5k
views
How to think about model categories?
I've read about model categories from an Appendix to one of Lurie's papers.
What are the examples of model categories? What should be my intuition about them?
E.g. I understand the typical examples ...
- 15.1k
6
votes
2
answers
1k
views
Higher vanishing cycles
The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-...
- 10.8k
5
votes
4
answers
826
views
$E_\infty$ spectrum corresponding to $\Bbb Z_p$
First of the questions about derived algebraic geometry from a noobie.
The way I understand it, every discrete ring $R$ corresponds to some ring spectrum whose $\pi_0$ is $R$. Now consider $p$-adic ...
- 15.1k