All Questions
Tagged with infinity-categories homological-algebra
31 questions
7
votes
2
answers
314
views
How to get an $E_\infty$-ring from a commutative differential graded ring?
I want to figure out the following question: How to get an $E_\infty$-ring from a commutative differential graded ring?
More precisely, let $\operatorname{cdga}$ be the ($1$-)category of cdgas, let $...
- 81
3
votes
1
answer
152
views
Homotopy coherent transformation and totalization
Let $C$ be the category of chain complexes over a field $F$ and $C^\prime$ be the subcategory of chain complexes with zero differentials. If $X:I\to C$ is a functor, there is an induced "homology&...
- 410
6
votes
1
answer
309
views
Comparing the Stacks Project Homotopy limit with limits in the $\infty$-category
In the Stacks project Tag 08TC, there is a definition of a homotopy limit in a derived category, and I expect it to compare with a limit in the $\infty$-categorical enhancement. I guess this is also ...
- 507
3
votes
2
answers
241
views
Adjunctions and inverse limits of derived categories
Consider a tower $\dots\to A_{2}\to A_{1}$ of rings. This gives rise to a diagram $\mathbb{N}^{\text{op}}\to\text{Cat}_{\infty}$ of $\infty$-categories (confusing $\mathbb{N}^{\text{op}}$ with its ...
- 507
7
votes
0
answers
273
views
Homotopy theory of differential objects
In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
- 6,025
2
votes
0
answers
310
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
956
views
Reference request: infinity categories for the commutive algebraist/algebraic geometer
In a survey article Algebraic geometry in mixed characteristic, B. Bhatt writes
For instance, given a commutative ring $R$ with a finitely generated ideal $I$,
the assignment carrying $R$ to the $\...
- 785
8
votes
2
answers
897
views
Derived functors out of an unbounded derived $\infty$-category
Let $\mathcal A$ be an abelian category. In this lecture, Thomas Nikolaus
Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\...
- 1,217
7
votes
1
answer
812
views
How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable?
Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module ...
3
votes
0
answers
170
views
Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?
Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...
- 2,152
9
votes
1
answer
615
views
Any news about equivalences of periodic triangulated or $\infty$-categories?
There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted ...
- 18.4k
29
votes
1
answer
4k
views
Why stable $\infty$-categories?
I begin by saying that while I understand what a triangulated / derived category is pretty well, I know nothing about Higher Algebra stuff and not even $\infty$-categories.
I've heard some people say ...
- 711
11
votes
2
answers
470
views
Is the $\infty$-category $N_{dg}(\mathrm{Ch}(\mathcal{A}))$ presentable?
(See Jacob Lurie's "Higher Algebra", section 1.3.5 for context.)
Let $\mathcal{A}$ be a Grothendieck abelian category. Then the stable $\infty$-category $\mathcal{D}(\mathcal{A})$ is a ...
- 111
10
votes
1
answer
1k
views
Functorial kernel in derived category
By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in these notes, it is said that
In the ...
- 4,408
9
votes
1
answer
683
views
Two $\infty$-categories of chain complexes
In the literature, I've mostly seen two quasicategories coming from $\text{Ch}_R$:
By considering $\text{Ch}_R$ with weak equivalences $\mathcal W = \text{quasi-isomorphisms}$, we can consider its ...
- 527
3
votes
0
answers
332
views
$E_{\infty}$-algebras à la Lurie
Let $D(\mathbb{F}_p)$ and $\mathcal{D}(\mathbb{F}_p)$ be the derived category and derived infinity-category of cochain complexes of $\mathbb{F}_p$-vector spaces.
If $A$ is a sheaf of cdgas over $\...
- 71
1
vote
2
answers
723
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
4
votes
0
answers
104
views
Adjunctions between $\mathcal{A}_{\infty}$-categories
Is there a theory of homotopy coherent adjunctions between $\mathcal{A}_{\infty}$-categories? By this I mean at least a definition of what an adjunction is and a construction of the corresponding ...
- 101
2
votes
0
answers
303
views
L-infinity algebra of deformations of an L-infinity algebra?
From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC ...
- 784
6
votes
0
answers
148
views
Hochschild cohomology of the $A_\infty$-category of paths
I would like to describe the Hochschild cohomology (in the sense of $A_\infty$-categories) of the following $A_\infty$-category associated to a topological space $X$:
It has points of $X$ as objects.
...
- 101
2
votes
0
answers
239
views
Reference Request: A "Chevalley-Eilenberg"-style formulation of the $L_\infty$ algebra minimal model theorem?
The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...
- 784
8
votes
1
answer
615
views
Functorial construction of ("pre"-)spectral sequences? (Or - what is the "higher structure" underlying spectral sequences?)
Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the ...
- 7,789
9
votes
0
answers
506
views
Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
- 7,789
4
votes
0
answers
86
views
BV on cyclic cohomology
Suppose that A is an associative algebra, but not Frobenius. Does the Cyclic Cohomology of A have a structure of a BV algebra?
All results I could find first identify A with its dual but I am asking ...
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:
$$\...
- 7,789
10
votes
5
answers
1k
views
Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?
When reading about homotopy algebras (e.g. $L_\infty$-algebras, $A_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f_1$ is a quasi-isomorphism.
Recall/Example (...
- 103
13
votes
2
answers
2k
views
teaching higher algebra
Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The kind of ...
- 534
5
votes
1
answer
915
views
Doing some homological algebra in triangulated categories
It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} \...
- 40.2k
9
votes
1
answer
584
views
Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?
My question is:
Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by $D^b_\infty\text{...
- 1,589
4
votes
0
answers
685
views
Unbounded derived category that is not left-complete
Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...
- 805
23
votes
3
answers
4k
views
how to make the category of chain complexes into an $\infty$-category
I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.
Has anyone ever written down ...
- 1,889