All Questions
Tagged with derived-categories homotopy-theory
28
questions
3
votes
0
answers
80
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
343
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
309
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 ...
- 5,038
10
votes
1
answer
1k
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
665
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
492
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 ...
- 5,780
3
votes
0
answers
125
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
292
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 ...
- 1,951
10
votes
1
answer
1k
views
Derived base change in étale cohomology
Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
- 27.9k
10
votes
1
answer
847
views
$\infty$-categorical understanding of Bridgeland stability?
On triangulated categories we have a notion of Bridgeland stability conditions.
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
- 103
9
votes
0
answers
316
views
Dualizable objects in homotopy category of chain complexes
The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that:
Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...
user95283
3
votes
1
answer
445
views
Spectral sequence for tensor product of complexes
Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X_{\rm ét}$.
I want to compute the hypercohomology:
$$\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 ...
- 5,953
35
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 ...
- 10k
4
votes
0
answers
72
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