Questions tagged [a-infinity-algebras]
For questions about $A_\infty$-algebras as introduced by Stasheff in 1963 and related structures.
12 questions
21
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3
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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
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2
answers
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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 $...
- 21k
33
votes
8
answers
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triangulated vs. dg/A-infinity
Someone recently said "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions".
I have a rough idea why this is true ("don't ...
- 12.8k
16
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6
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Is an A-infinity thing the same the same as strict thing viewed through a homotopy equivalence?
If I have a topological monoid ($X,\mu$), a space $Y$ and a homotopy equivalence $f,g$ between them, then $Y$ has the structure of an $A_\infty$ space defined 'pointwise' by
$$ y_1 * y_2 := g \left(\...
- 461
11
votes
1
answer
690
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Are $(\infty,1)$-categories $A_\infty$ categories?
Let $X$ be a set. One can define a non-symmetric colored version of the non-unital $A_\infty$ operad as follows. The set of colors is the set of ordered pairs in $X$. Let $(x_1,y_1),\dots,(x_n,y_n)$...
- 54.6k
11
votes
1
answer
1k
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Why Grothendieck's Homotopy Hypothesis is so difficult?
Recall that Grothendieck's Homotopy Hypothesis states that the homotopy category of weak globular $n$-groupoids (as in https://arxiv.org/pdf/1009.2331.pdf) is equivalent to the category of homotopy $n$...
- 2,227
11
votes
1
answer
759
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Model structure on the category of small $A_\infty$ categories, hocolims.
I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in ...
- 187
10
votes
1
answer
657
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Tensor products of $\infty$-algebras over operads
Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to ...
- 40.3k
7
votes
3
answers
3k
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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.
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- 21k
5
votes
1
answer
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Homotopic morphisms between curved A-infinity algebras
I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in ...
- 460
4
votes
1
answer
244
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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
2
votes
0
answers
92
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Monoidal structure on left dg-modules over a brace algebra
Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
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