6
votes
2answers
355 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 ...
2
votes
2answers
357 views

Homotopic monoids and $A_\infty$ spaces

Informally, an $A_\infty$-space is a monoid whose laws are only satisfied up to homotopy. Let’s define now what I will call a "homotopic monoid" to be a space $M$ together with a point $e\in{}M$ and ...
6
votes
1answer
266 views

Does there exist a model of chains on oriented manifolds with both a strict intersection pairing and strict functoriality for closed embeddings?

Let $M$ be a smooth oriented $n$-dimensional manifold. My favorite model of $\operatorname{Chains}_\bullet(M) \otimes \mathbb R$ is the space of smooth compactly-supported de Rham forms on $M$, ...
11
votes
4answers
831 views

Is an A-infinity thing the same the same as strict thing viewed through a homotpy equivalence?

If I have a topological monoid ($X,\mu$), a space $Y$ and a homotopy equivalence $f,g$ between them, then $Y$ has the strucutre of an $A_\infty$ space defined 'pointwise' by $$ y_1 * y_2 := g ...
10
votes
2answers
803 views

A-infinity tensor categories

My question is rather simple: What is the correct notion of a monoidal A-infinity category C? Or is there any reference where such a notion is explained?
3
votes
1answer
580 views

$A_{\infty}$ structure of (co)homology of a space

Let $X$ be a topological space, and $Homeo(X)$ the group of self-homeomorphisms of $X$. (1) What is the exact meaning of: $H^*(X)$ is a an $A_\infty$-module over $Homeo(X)$? (2) Does $H_*(X)$ also ...
19
votes
8answers
2k views

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 ...
7
votes
2answers
842 views

What's the sense in which A_\infty algebras are “deformable”?

I realise this is a very vague question! I've heard people say that A∞ algebras are the right homotopy-theoretic generalization of usual associative algebras, because you can deform them. What ...