All Questions
Tagged with a-infinity-algebras differential-graded-algebras
19 questions
6
votes
2
answers
570
views
Is the exterior algebra intrinsically formal?
Following 4.6 and 4.7 of this paper by Seidel and Thomas, a graded algebra $A$ is called intrinsically formal if any two dgas with cohomology $A$ are quasi-isomorphic. There is a sufficient condition ...
- 673
3
votes
0
answers
365
views
Construction of derived Quot schemes
I am studying the construction of derived Quot schemes in the paper Borisov, Katzarkov, and Sheshmani - “Shifted symplectic structures on derived Quot-stacks”.
Derived quot stacks are constructed from ...
- 479
8
votes
2
answers
417
views
Conceptual explanation for the sign in front of some binary operations
In several situations, I've seen that given a binary operation on a graded module $m:A\otimes A\to A$, a new operation $M(x,y)=(-1)^{|x|}m(x,y)$ is defined so that it satisfies some properties.
One ...
- 499
1
vote
0
answers
98
views
Construct $A_\infty$ bimodules maps from dg-maps
Let $ A $ be a dg-algebra. Let $U,V,W$ and $Z$ be dg-bimodules over $A$-$A$. Suppose I have cofibrant replacements $\pi_U : Up \rightarrow U$ (as right dg-module) and $\pi_W : pW \rightarrow W$ (as ...
- 71
3
votes
2
answers
191
views
Two definitions of minimal models
Is there any relationship between both definitions of minimal models? (the couple of definitions I know are the one mentioned in Lefèvre's thesis, in the sense that the differential is zero, and the ...
- 795
6
votes
1
answer
224
views
Is the existence of $A_{\infty}$-inverse a consequence of Homotopy Transfer Theorem?
Let $k$ be a field of characteristic $0$ and $(A,d_A)$, $(B,d_B)$ be two differential graded (dg) algebras over $k$. Let $f: A\to B$ be a closed degree $0$ map of dg-algebras and $g: B\to A$ be a map ...
- 9,019
11
votes
1
answer
506
views
If C is a cocomplete coalgebra, then $\psi:C\rightarrow B\Omega C$ is a filtered quasi-isomorphism
I am reading the PhD thesis thesis of Kenji Lefèvre-Hasegawa and the corresponding errata by Bernhard Keller, my question is about the first error found in the thesis. Lemma 1.3.2.3 c states 'the ...
- 795
11
votes
1
answer
534
views
On the coalgebraic homotopy transfer theorem
Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A)$...
- 40.3k
4
votes
1
answer
664
views
Homology of bar complex vs homology of indecomposables
$\require{AMScd}$
Background: This question is about the bar and cobar constructions, and their relationship with the indecomposables of a dg-algebra. A brief summary of the bar and cobar ...
- 1,231
3
votes
0
answers
106
views
Explicit description of periodic map $S : HC_{i} \to HC_{i-2}$ for dg and $A_\infty$ algebras
Let $A$ some associative unital $k$-algebra, let $HC_*(A)$ is cyclic homology of $A$ and $HH_*(A)$ is hochschild homology of $A$. Then we have Connes exact sequence:
$$ ... \xrightarrow[]{} HH_n(A) \...
- 1,307
15
votes
1
answer
558
views
Defining Massey products as transgressions
Let $A$ be a dg algebra, and $x, z \in A$ cocycles. Let's consider the maps
$$ A \to A \oplus A \to A$$
given by $y \mapsto (xy,yz)$ and $(u,v) \mapsto uz-xv$, respectively. We think of this as ...
- 40.3k
4
votes
0
answers
162
views
Which dg-algebras have minimal model which is $A_{fin}$?
$A_{fin}$ algebra it is $A_\infty$ algebra with $m_n = 0$ for $n >> 0$ and $A^i = 0$ for $|i| >> 0$.
Suppose that we have (compact) dg-algebra $A$, we can build $A_\infty$ minimal model ...
- 1,307
3
votes
0
answers
476
views
(co)chain homotopy of dg algebras ($A_{\infty}$ algebras) and other notion of homotopy
In this question I will work over a field of char. $0$. Let $f,g\: : \: A\to B$ be dg algebras morphism between two cochain (commutative) dg algebras $A,B$ (assume positively graded). Let $h$ be a ...
- 1,424
8
votes
1
answer
674
views
Frobenius $A_{\infty}$-bialgebras?
Recall that a finite dimensional associative algebra $A$ over a field $k$ is called a symmetric Frobenius algebra (sometimes called "closed" Frobenius algebra) if its equipped with a symmetric non ...
- 2,035
13
votes
2
answers
1k
views
The cohomology plus what characterizes the rational homotopy type?
For simplicity let me work only with connected and simply connected spaces. "Space" will mean a space of this type.
A space is rational if its homotopy groups are rational vector spaces (...
- 27.5k
9
votes
2
answers
3k
views
Homotopy Transfer Theorem for Differential Graded Associative Algebras
As in Algebra+Homotopy=Operad by Bruno Vallette, let $A$ with multiplication $\nu$ be a differential graded associative algebra equipped with degree +1 map $h$ and let $H$ be a chain complex such that ...
- 996
2
votes
1
answer
1k
views
Associated graded of a filtration of a tensor product
I'm trying to understand a part of the PhD thesis of Kenji Lefèvre-Hasegawa (e.g. available here). My question is about the proof of Lemma 1.3.2.3b stating:
Remarquons que nous avons un ...
- 2,450
21
votes
3
answers
2k
views
Formality of classifying spaces
Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on ...
- 6,162
16
votes
2
answers
3k
views
Smooth dg algebras (and perfect dg modules and compact dg modules)
Katzarkov-Kontsevich-Pantev define a smooth dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is perfect if the functor $...
- 21k