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created: 2023-03-31T19:28
updated: 2023-03-31T19:30
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date: 2022-02-07 23:10
modification date: Sunday 13th February 2022 20:32:35
title: A_infty
aliases: [A_infty algebra, A_infty space, A_infty category]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #geomtop/symplectic-topology #geomtop/Floer-theory 
- Refs:
	- <https://arxiv.org/abs/math/0601185>
- Links:
	- [L infty algebra](Unsorted/L%20infty%20algebra.md)
	- [DGA](Unsorted/DGA.md)
	- [CY](Unsorted/Calabi-Yau.md)
	- [cobar construction](Unsorted/bar%20construction.md)
	- [L infty algebra](Unsorted/L%20infty%20algebra.md)
	- [dg manifold](dg%20manifold.md)
	- [Calabi-Yau](Calabi-Yau.md) $A_\infty\dash$structures are studied in the context of [open topological string theory](open%20topological%20string%20theory).


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# $A_\infty$ structures

Idea: associativity only up to higher coherent homotopy.
Naturally present on the homology complex of any [DGA](Unsorted/DGA.md).

![](attachments/2023-03-31-85.png)
![](attachments/2023-03-31-86.png)
![](attachments/2023-03-31-87.png)

![](attachments/2023-03-31-88.png)


Definitions:
![](attachments/2023-03-31-89.png)
![](attachments/2023-03-31-90.png)
![](attachments/2023-03-31-91.png)
Examples:

- $H_{\dR}(\wait)$
- - $\HoH(\wait)$
- Any [loop space](Unsorted/loop%20space.md).
  
![](attachments/2023-03-31-92.png)
![](attachments/2023-03-31-93.png)

# Coherence

![](attachments/2023-03-31-94.png)
# A_infty algebra

![](attachments/2023-03-31-95.png)
Here

- First relation $m_1^2=0$ makes $A$ a chain complex
- Second relation makes multiplication a chain morphism
- Third relation makes $m_2$ associative up to chain homotopy
- Higher relations impose homotopy coherence


See [Gerstenhaber bracket](Gerstenhaber%20bracket.md), [coalgebra](Unsorted/coalgebra.md), [Maurer-Cartan](Unsorted/Maurer-Cartan.md) equations.
![](attachments/2023-03-31-96.png)

# pre-CY structures

Somehow give rise to [TQFTs](Unsorted/TQFT.md).
![](attachments/2023-03-31-97.png)


# A_infty categories
![](attachments/2023-03-31-98.png)
![](attachments/2023-03-31-99.png)
![](attachments/2023-03-31-100.png)![](attachments/2023-03-31-101.png)

![](2023-03-31-117.png)

# Twisted complexes

![](attachments/2023-03-31-102.png)
![](attachments/2023-03-31-103.png)

![](2023-03-31-118.png)

![](2023-03-31-119.png)
# Bimodules

Related to [tilting modulues](tilting%20modulues)
![](attachments/2023-03-31-104.png)