All Questions
Tagged with a-infinity-algebras reference-request
16 questions
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
18
votes
1
answer
991
views
Higher homotopy algebraic structure on the homology of an operad
Given a DGA $A$, then by standard techniques such as homological perturbation theory, the ring structure on the homology $H(A)$ extends to a minimal $A_\infty$-algebra structure such that $H(A)$ is ...
- 6,229
13
votes
5
answers
2k
views
$A_\infty$-categories basic reference
Can anyone provide me with a basic reference on $A_\infty$ categories?
user16974
12
votes
2
answers
799
views
Reference for functors in Kadeishvili's C_\infty paper
In his paper Cohomology $C_\infty$-algebra and rational homotopy type, Tornike Kadeishvili describes how the rational cohomology of a simply-connected space carries the structure of a $C_\infty$-...
- 35.9k
7
votes
3
answers
3k
views
Hochschild cohomology and A-infinity deformations
When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.
...
- 21k
7
votes
0
answers
244
views
Operad-free proofs of rectification of homotopy ($A_\infty/L_\infty$) algebras?
If (say) $L$ is an $L_\infty$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $L_\infty$-algebra $L'$ which is a differential graded Lie algebra: all ternary ...
- 784
5
votes
1
answer
367
views
Homotopy invariant structure: Stasheff versus Segal
To describe homotopy invariant algebraic structures on spaces, there are different approaches.
The Stasheff / Boardman–Vogt / May approach, where operations and equations are replaced by spaces of ...
- 1,243
5
votes
1
answer
675
views
Resolution of a module as an $A_\infty$ module over resolution of an algebra
The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...
- 1,545
5
votes
0
answers
303
views
Recovering an A-infinity structure on an Ext-algebra from a quiver presentation
Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules.
According to [Keller: A-infinity algebras in ...
- 2,450
5
votes
0
answers
151
views
$A_\infty$ structure on sum of twists of structure sheaf
Fix $n$ and let $P^n$ be projective $n$-space. Let $S = k[x_0, \dots, x_n]$. Set $A^0 = \bigoplus_{d \ge 0} H^0(P^n, \mathcal{O}(d))$ and $A^n = \bigoplus_{d < -n} H^n(P^n, \mathcal{O}(d))$.
I ...
- 10.7k
4
votes
1
answer
244
views
Are these two natural $A_\infty$-structures on the realization of a cosimplicial commutative algebra isomorphic?
Given a cosimplicial commutative algebra $A^\bullet$ over a field of characteristic zero, there are two ways of producing an $A_\infty$-structure on its realization $|A^\bullet| := \int^\Delta C^*(\...
- 4,709
4
votes
0
answers
126
views
Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
- 489
4
votes
0
answers
225
views
Natural transformations of $A_\infty$-functors (between dg-categories) are "directed homotopies" (reference?)
Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to \...
- 2,079
3
votes
1
answer
167
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
0
answers
261
views
On the Hochschild cohomology of the minimal model of an $A_\infty$ algebra
Suppose $(A, (\mu_k))$ is a (curved) $A_\infty$ algebra, and let $(\tilde A, (\tilde\mu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(\tilde A)$. (It may ...
- 2,789
1
vote
0
answers
79
views
Reference request for showing open(resp. closed) string field theory has A-infinity(resp. L-infinity) algebra structure
I've now begun to study about the relationship between open(resp. closed) string field theory and A-infinity(resp. L-infinity) algebra structure.
For the A-infinity case, I'd already heard that the ...
- 285