All Questions
Tagged with homological-algebra higher-category-theory
41 questions
3
votes
1
answer
163
views
Theory of $n$-truncated $A_\infty$ categories/functors?
One can define certain higher categorical truncations. For example, a discussion from the quasi-categories point of view can be found in HTT 5.5.6.
On the other hand, as a model of linear $\infty$-...
- 169
3
votes
1
answer
359
views
Is the adjunction between spaces and chain complexes monadic?
Consider the adjunction of $\infty$-categories $\mathbb{Z}[-]: \mathcal{Spaces} \rightleftarrows \text{Ch}_{\ge 0}(\mathbb{Z}): |-|$ where the left adjoint takes a space to its singular chain complex ...
- 423
4
votes
0
answers
238
views
Derived functors from localization vs animation
I got a bit confused with the derived functors getting from the localization and the animation. More specifically, let $\mathcal{A}$ be an abelian category generated by compact projective objects $\...
- 255
4
votes
0
answers
456
views
An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?
Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...
- 889
5
votes
0
answers
204
views
Preservation of (co)limits under taking derived categories
Let $R$ be a commutative ring. Let $\{A_i\}_{i \in I}$ be a diagram of $R$-linear $1$-categories, indexed by a finite poset $I$. (If this matters, assume that the $A_i$ have finitely many objects).
...
- 373
2
votes
0
answers
171
views
Motivation for working with augmented objects in homological or higher algebra
I would like to understand if there is deeper reason/motivation behind
augmentations in homological algebra. Recall classically in homology
if there is a complex of free $R$-modules ($R commutative ...
- 6,018
5
votes
0
answers
161
views
Cohomology of a countable directed union of groups
It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a ...
- 1,638
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
2
votes
0
answers
193
views
When does Tate spectral sequence degenerate at $E_2$?
For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence
$$
E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
- 519
7
votes
0
answers
302
views
The role of spectral Lie algebras and twisting for operads in spectra
In the theory of differential graded (co)operads, the notion of twisting is ubiquitous. The fundamental notion is the twisting map from a cooperad $C$ to an operad $P$. It is defined as a Maurer-...
- 5,839
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
25
votes
1
answer
1k
views
What are Koszul dualities?
I am bewildered by the number of things I've heard referred to as "Koszul duality", and I would like to sort it out. At various different times, I believe I've seen any of the following ...
- 63.9k
2
votes
0
answers
194
views
The derived $\infty$-category of sheaves on a site is closed symmetric monoidal
Let $X$ be a quasicompact semiseparated scheme. I am trying to recover the (closed) symmetric monoidal structure on $\mathcal{D}(\mathrm{QCoh}(X))$, the derived $\infty$-category of quasicoherent ...
- 617
1
vote
1
answer
360
views
Computing Ext groups in a functor stable $\infty$-category
Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. ...
- 617
2
votes
0
answers
184
views
Tensor product and cohomology of dg categories
Let $\mathcal{C}$ and $\mathcal{D}$ be dg categories over a field $k$ of characteristic zero. Then one can form their tensor product $\mathcal{C} \otimes \mathcal{D}$: the objects of the tensor ...
- 937
6
votes
1
answer
283
views
Perfect $\mathbb Z_\ell$-modules
Disclaimer : I asked this question on Maths.StackExchange 20 days ago (and started a bounty) here but got no answer, so I'm asking it here now (with no modification).
$\newcommand{\l}{\ell} \...
- 15.8k
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
3
votes
0
answers
137
views
Bar-cobar for topological spaces
Let $(A, \{m_k\}_{k \geq 1} )$ be an $A_{\infty}$ algebra over a field $k$. Recall that this is the data of a $\mathbb{Z}$-graded $k$-vector space, along with a collection of $k$-nary operations which ...
- 937
3
votes
0
answers
371
views
Fibered category vs indexed category
In Benabou's paper Fibered categories and the foundations of naive category theory, it is mentioned (end of page 31) that 'an indexed category is just a presentation of a fibered category.'
It seems ...
- 2,789
1
vote
1
answer
130
views
'Quotient' of indexed categories and pushforward congruence relations
If $R\subset C$ is a vector subspace of $C$, and $F: C \to D$ is a linear map, then we have a natural linear map $C/R\to D/F(R)$. I was wondering if this can be also generalized to categorical ...
- 2,789
1
vote
0
answers
165
views
Translate a construction into categorical languages
Let $\mathscr C$ be a category whose objects are sets.
Let $\mathscr V$ be another category. It seems that the usual composition in $\mathscr V$ can be somehow 'twisted' by $\mathscr C$. I notice that ...
- 2,789
2
votes
0
answers
139
views
$\omega$-categorical algebra
Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be ...
- 517
3
votes
0
answers
208
views
Cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$
Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group.
(For example, given a short exact sequence
$$
1 \to BG_2 \to \mathbb{G} \to G_1 \to 1
$$
and the fiber sequence:
$$
B^2G_2 ...
- 10.5k
2
votes
2
answers
212
views
Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step ...
- 7,789
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
3
votes
0
answers
61
views
Can one relate $K_0$ of an $A_\infty$-category $\mathcal A$ to $K_0(Fun_{A_\infty}(\mathcal A, \mathcal A))$?
For an $A_\infty$-category $\mathcal A$, one defines the group $K_0(\mathcal A)$ by
$$K_0(\mathcal A) := \mathbb Z \operatorname{Ob} \operatorname{Tw} \mathcal A / \left<[A]+[B]-[C]\right>$$
...
- 1,893
7
votes
0
answers
268
views
Identifying and reconstructing the derived category from its auto-equivalences
Background: Given a smooth irreducible algebraic variety $Y$ with $\omega_Y$ or $\omega_Y^{-1}$ ample. Then Bondal-Orlov theorem states that if there exists any other smooth algebraic variety $Y'$ ...
- 1,981
2
votes
0
answers
143
views
homotopy quotient categories [closed]
(Trying to rephrase an earlier question)
In topology, a continuous map $f: X \to Y$ has a ``homotopy cofiber" $Cf$ included in a cofiber sequence
$$
X \to Y \to Cf \to \Sigma X \to \Sigma Y \to \...
6
votes
1
answer
274
views
Switching left and right adjoints in recollement situations
In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors
$$
...
- 6,777
6
votes
2
answers
337
views
Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?
Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
RHom(C,...
- 9,019
9
votes
1
answer
550
views
derived categories as presentable DG-categories
Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and ...
- 5,562
1
vote
0
answers
82
views
Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones
This question is strictly related to this other one.
Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let
(source: presheaf.com)
be a diagram in $Z^0(\mathcal A)$, ...
- 2,079
5
votes
2
answers
501
views
Homotopy factorization of morphisms of chain complexes
This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
- 6,777
3
votes
1
answer
381
views
Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?
Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
- 9,019
15
votes
4
answers
2k
views
What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?
Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence ...
- 9,019
9
votes
1
answer
370
views
Analogue of cyclic homology for e_n-algebras?
Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the "...
- 743
18
votes
3
answers
2k
views
A 2-category of chain complexes, chain maps, and chain homotopies?
First-time here... I hope my question isn't silly or anything... anyway...
Consider the category of chain complexes and chain maps. We can also define chain homotopies between chain maps. Does this ...
- 183
0
votes
0
answers
349
views
cokernel for $L_\infty$-algebra morphisms
As I have asked a wrong question previously, I edited a bit.
It is obvious that the cokernel construction does not work well for the category of Lie algebras. The cokernel exists only for a normal ...
- 1,271
23
votes
3
answers
3k
views
Homology theory constructed in a homotopy-invariant way
Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
- 3,033
8
votes
1
answer
628
views
Is there a "derived" Free $P$-algebra functor for an operad $P$?
Recall that an operad (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S_n$, with maps $P(...
- 54.6k
4
votes
3
answers
666
views
Chain Complexes and Linear Infinity-Categories
A statement I heard recently is that "chain complexes are the same thing as strict linear $\infty$-categories". Can someone explain how to see this?
- 2,806