Homotopy associative algebra

In mathematics, an algebra such as $$(\R,+,\cdot)$$ has multiplication $$\cdot$$ whose associativity is well-defined on the nose. This means for any real numbers $$a,b,c\in \R$$ we have
 * $$a\cdot(b\cdot c) - (a\cdot b)\cdot c = 0$$.

But, there are algebras $$R$$ which are not necessarily associative, meaning if $$a,b,c\in R$$ then
 * $$a\cdot(b\cdot c) - (a\cdot b)\cdot c \neq 0$$

in general. There is a notion of algebras, called $$A_\infty$$-algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative. This means although we get something which looks like the second equation, the one of inequality, we actually get equality after "compressing" the information in the algebra.

The study of $$A_\infty$$-algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algebras through a differential graded algebra with a multiplication operation and a series of higher homotopies giving the failure for the multiplication to be associative. Loosely, an $$A_\infty$$-algebra $$(A^\bullet, m_i)$$ is a $$\Z$$-graded vector space over a field $$k$$ with a series of operations $$m_i$$ on the $$i$$-th tensor powers of $$A^\bullet$$. The $$m_1$$ corresponds to a chain complex differential, $$m_2$$ is the multiplication map, and the higher $$m_i$$ are a measure of the failure of associativity of the $$m_2$$. When looking at the underlying cohomology algebra $$H(A^\bullet, m_1)$$, the map $$m_2$$ should be an associative map. Then, these higher maps $$m_3,m_4,\ldots$$ should be interpreted as higher homotopies, where $$m_3$$ is the failure of $$m_2$$ to be associative, $$m_4$$ is the failure for $$m_3$$ to be higher associative, and so forth. Their structure was originally discovered by Jim Stasheff while studying A∞-spaces, but this was interpreted as a purely algebraic structure later on. These are spaces equipped with maps that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so forth.

They are ubiquitous in homological mirror symmetry because of their necessity in defining the structure of the Fukaya category of D-branes on a Calabi–Yau manifold who have only a homotopy associative structure.

Definition
For a fixed field $$k$$ an $$A_\infty$$-algebra is a $$\Z$$-graded vector space
 * $$A = \bigoplus_{p \in \Z}A^p$$

such that for $$d \geq 1$$ there exist degree $$2 - d$$, $$k$$-linear maps
 * $$m_d\colon (A^{\bullet})^{\otimes d} \to A^\bullet$$

which satisfy a coherence condition:
 * $$\sum_{

\begin{matrix} 1 \leq p \leq d \\ 0 \leq q \leq d-p \end{matrix} }(-1)^\alpha m_{d-p+1}(a_d, \ldots, a_{p+q+1}, m_p(a_{p+q},\ldots, a_{q+1}), a_q,\ldots,a_1) = 0$$, where $$\alpha = (-1)^{\text{deg}(a_1) + \cdots + \deg(a_q) - q}$$.

Understanding the coherence conditions
The coherence conditions are easy to write down for low degrees pgs 583–584.

d=1
For $$d = 1$$ this is the condition that
 * $$m_1(m_1(a_1)) = 0 $$,

since $$1 \leq p \leq 1$$ giving $$p = 1 $$ and $$0 \leq q \leq d - 1 $$. These two inequalities force $$m_{d-p+1} = m_{1} $$ in the coherence condition, hence the only input of it is from $$m_1(a_1) $$. Therefore $$m_1 $$ represents a differential.

d=2
Unpacking the coherence condition for $$d=2 $$ gives the degree $$0 $$ map $$m_2 $$. In the sum there are the inequalities
 * $$\begin{matrix}

1 \leq p \leq 2 \\ 0 \leq q \leq 2-p \end{matrix} $$ of indices giving $$(p,q) $$ equal to $$(1,0),(1,1),(2,0) $$. Unpacking the coherence sum gives the relation
 * $$m_2(a_2, m_1(a_1)) + (-1)^{\deg(a_1) - 1}m_2(m_1(a_2), a_1) + m_1(m_2(a_1,a_2)) = 0 $$,

which when rewritten with
 * $$(-1)^{\deg a}m_1(a) = d(a) $$ and $$(-1)^{\deg a_1}m_2(a_2,a_1) = a_2\cdot a_1 $$

as the differential and multiplication, it is
 * $$d(a_2\cdot a_1) = (-1)^{\deg(a_1)}d(a_2)\cdot a_1 +a_2\cdot d(a_1) $$,

which is the Leibniz rule for differential graded algebras.

d=3
In this degree the associativity structure comes to light. Note if $$m_3=0 $$ then there is a differential graded algebra structure, which becomes transparent after expanding out the coherence condition and multiplying by an appropriate factor of $$(-1)^k$$, the coherence condition reads something like
 * $$\begin{align}

m_2(m_2(a\otimes b)\otimes c) - m_2(a\otimes m_2(b\otimes c)) =& \pm m_3(m_1(a)\otimes b\otimes c) \\ & \pm m_3(a\otimes m_1(b)\otimes c) \\ & \pm m_3(a\otimes b\otimes m_1(c)) \\ & \pm m_1(m_3(a\otimes b \otimes c)). \end{align}$$ Notice that the left hand side of the equation is the failure for $$m_2$$ to be an associative algebra on the nose. One of the inputs for the first three $$m_3$$ maps are coboundaries since $$m_1$$ is the differential, so on the cohomology algebra $$(H^*(A^\bullet, m_1),[m_2])$$ these elements would all vanish since $$m_1(a) = m_1(b) = m_1(c) = 0$$. This includes the final term $$m_1(m_3(a\otimes b \otimes c))$$ since it is also a coboundary, giving a zero element in the cohomology algebra. From these relations we can interpret the $$m_3$$ map as a failure for the associativity of $$m_2$$, meaning it is associative only up to homotopy.

d=4 and higher order terms
Moreover, the higher order terms, for $$d \geq 4$$, the coherent conditions give many different terms combining a string of consecutive $$a_{p+i},\ldots,a_{p+1}$$ into some $$m_p$$ and inserting that term into an $$m_{d-p+1}$$ along with the rest of the $$a_j$$'s in the elements $$a_d,\ldots,a_1$$. When combining the $$m_1$$ terms, there is a part of the coherence condition which reads similarly to the right hand side of $$d=3$$, namely, there are terms
 * $$\begin{align}

&\pm m_d(a_d,\ldots, a_2,m_1(a_1)) \\ &\pm \cdots \\ &\pm m_d(m_1(a_d),a_{d-1},\ldots, a_1) \\ &\pm m_1(m_d(a_d,\ldots, a_1)). \end{align}$$ In degree $$d=4$$ the other terms can be written out as
 * $$\begin{align}

& \pm m_3(m_2(a_4,a_3), a_2,a_1) \\ & \pm m_3(a_4,m_2(a_3,a_2),a_1) \\ & \pm m_3(a_4,a_3,m_2(a_2,a_1)) \\ & \pm m_2(m_3(a_4,a_3,a_2), a_1) \\ & \pm m_2(a_4,m_3(a_3,a_2,a_1)), \end{align}$$ showing how elements in the image of $$m_3$$ and $$m_2$$ interact. This means the homotopy of elements, including one that's in the image of $$m_2$$ minus the multiplication of elements where one is a homotopy input, differ by a boundary. For higher order $$d>4$$, these middle terms can be seen how the middle maps $$m_2,\ldots, m_{d-1}$$ behave with respect to terms coming from the image of another higher homotopy map.

Diagrammatic interpretation of axioms
There is a nice diagrammatic formalism of algebras which is described in Algebra+Homotopy=Operad explaining how to visually think about this higher homotopies. This intuition is encapsulated with the discussion above algebraically, but it is useful to visualize it as well.

Associative algebras
Every associative algebra $$(A,\cdot)$$ has an $$A_\infty$$-infinity structure by defining $$m_2(a,b) = a\cdot b$$ and $$m_i = 0$$ for $$i \neq 2$$. Hence $$A_\infty$$-algebras generalize associative algebras.

Differential graded algebras
Every differential graded algebra $$(A^\bullet, d)$$ has a canonical structure as an $$A_\infty$$-algebra where $$m_1 = d$$ and $$m_2$$ is the multiplication map. All other higher maps $$m_i$$ are equal to $$0$$. Using the structure theorem for minimal models, there is a canonical $$A_\infty$$-structure on the graded cohomology algebra $$HA^\bullet$$ which preserves the quasi-isomorphism structure of the original differential graded algebra. One common example of such dga's comes from the Koszul algebra arising from a regular sequence. This is an important result because it helps pave the way for the equivalence of homotopy categories"$\text{Ho}(\text{dga}) \simeq \text{Ho}(A_\infty\text{-alg})$"of differential graded algebras and $$A_\infty$$-algebras.

Cochain algebras of H-spaces
One of the motivating examples of $$A_\infty$$-algebras comes from the study of H-spaces. Whenever a topological space $$X$$ is an H-space, its associated singular chain complex $$C_*(X)$$ has a canonical $$A_\infty$$-algebra structure from its structure as an H-space.

Example with infinitely many non-trivial mi
Consider the graded algebra $$V^\bullet = V_0\oplus V_1$$ over a field $$k$$ of characteristic $$0$$ where $$V_0$$ is spanned by the degree $$0$$ vectors $$v_1,v_2$$ and $$V_1$$ is spanned by the degree $$1$$ vector $$w$$. Even in this simple example there is a non-trivial $$A_\infty$$-structure which gives differentials in all possible degrees. This is partially due to the fact there is a degree $$1$$ vector, giving a degree $$k$$ vector space of rank $$1$$ in $$(V^\bullet)^{\otimes k}$$. Define the differential $$m_1$$ by
 * $$\begin{align}

m_1(v_0) = w \\ m_1(v_1) = w, \end{align}$$ and for $$d \geq 2$$
 * $$\begin{align}

m_d(v_1\otimes w^{\otimes k}\otimes v_1\otimes w^{\otimes (d-2)-k}) &=(-1)^ks_dv_1 & 0 \leq k \leq d-2 \\ m_d(v_1\otimes w^{\otimes (d-2)}\otimes v_2)&= s_{d+1}v_1 \\ m_d(v_1\otimes w^{\otimes (d-1)}) &= s_{d+1}w, \end{align}$$ where $$m_n = 0$$ on any map not listed above and $$s_n = (-1)^{(n-1)(n-2)/2}$$. In degree $$d=2$$, so for the multiplication map, we have $$\begin{align} m_2(v_1,v_1)&= -v_1 \\ m_2(v_1,v_2)&= v_1 \\ m_2(v_1,w)&= w. \end{align}$$ And in $$d=3$$ the above relations give
 * $$\begin{align}

m_3(v_1,v_1,w) &= v_1 \\ m_3(v_1,w,v_1) &= -v_1 \\ m_3(v_1,w,v_2) &= -v_1 \\ m_3(v_1,w,w) &= -w. \end{align}$$ When relating these equations to the failure for associativity, there exist non-zero terms. For example, the coherence conditions for $$v_1,v_2,w$$ will give a non-trivial example where associativity doesn't hold on the nose. Note that in the cohomology algebra $$H^*(V^\bullet, [m_2])$$ we have only the degree $$0$$ terms $$v_1,v_2$$ since $$w$$ is killed by the differential $$m_1$$.

Transfer of A ∞ structure
One of the key properties of $$A_\infty$$-algebras is their structure can be transferred to other algebraic objects given the correct hypotheses. An early rendition of this property was the following: Given an $$A_\infty$$-algebra $$A^\bullet$$ and a homotopy equivalence of complexes
 * $$f\colon B^\bullet \to A^\bullet$$,

then there is an $$A_\infty$$-algebra structure on $$B^\bullet$$ inherited from $$A^\bullet$$ and $$f$$ can be extended to a morphism of $$A_\infty$$-algebras. There are multiple theorems of this flavor with different hypotheses on $$B^\bullet$$ and $$f$$, some of which have stronger results, such as uniqueness up to homotopy for the structure on $$B^\bullet$$ and strictness on the map $$f$$.

Minimal models and Kadeishvili's theorem
One of the important structure theorems for $$A_\infty$$-algebras is the existence and uniqueness of minimal models – which are defined as $$A_\infty$$-algebras where the differential map $$m_1 = 0$$ is zero. Taking the cohomology algebra $$HA^\bullet$$ of an $$A_\infty$$-algebra $$A^\bullet$$ from the differential $$m_1$$, so as a graded algebra,
 * $$HA^\bullet = \frac{\text{ker}(m_1)}{m_1(A^\bullet)}$$,

with multiplication map $$[m_2]$$. It turns out this graded algebra can then canonically be equipped with an $$A_\infty$$-structure,
 * $$(HA^\bullet, 0, [m_2], m_3,m_4,\ldots)$$,

which is unique up-to quasi-isomorphisms of $$A_\infty$$-algebras. In fact, the statement is even stronger: there is a canonical $$A_\infty$$-morphism
 * $$(HA^\bullet, 0, [m_2], m_3,m_4,\ldots) \to A^\bullet$$,

which lifts the identity map of $$A^\bullet$$. Note these higher products are given by the Massey product.

Motivation
This theorem is very important for the study of differential graded algebras because they were originally introduced to study the homotopy theory of rings. Since the cohomology operation kills the homotopy information, and not every differential graded algebra is quasi-isomorphic to its cohomology algebra, information is lost by taking this operation. But, the minimal models let you recover the quasi-isomorphism class while still forgetting the differential. There is an analogous result for A∞-categories by Maxim Kontsevich and Yan Soibelman, giving an A∞-category structure on the cohomology category $$H^*(D^b_\infty(X))$$ of the dg-category consisting of cochain complexes of coherent sheaves on a non-singular variety $$X$$ over a field $$k$$ of characteristic $$0$$ and morphisms given by the total complex of the Cech bi-complex of the differential graded sheaf $$\mathcal{Hom}^\bullet(\mathcal{F}^\bullet,\mathcal{G}^\bullet)$$ pg 586-593. In this was, the degree $$k$$ morphisms in the category $$H^*(D^b_\infty(X))$$ are given by $$\text{Ext}(\mathcal{F}^\bullet,\mathcal{G}^\bullet)$$.

Applications
There are several applications of this theorem. In particular, given a dg-algebra, such as the de Rham algebra $$(\Omega^\bullet(X),d,\wedge)$$, or the Hochschild cohomology algebra, they can be equipped with an $$A_\infty$$-structure.

Massey structure from DGA's
Given a differential graded algebra $$(A^\bullet, d)$$ its minimal model as an $$A_\infty$$-algebra $$(HA^\bullet, 0, [m_2], m_3,m_4,\ldots)$$ is constructed using the Massey products. That is,
 * $$\begin{align}

m_3(x_3,x_2,x_1) &= \langle x_3,x_2,x_1 \rangle \\ m_4(x_4,x_3,x_2,x_1) &= \langle x_4, x_3,x_2,x_1 \rangle \\ &\cdots & \end{align}$$ It turns out that any $$A_\infty$$-algebra structure on $$HA^\bullet$$ is closely related to this construction. Given another $$A_\infty$$-structure on $$HA^\bullet$$ with maps $$m_i'$$, there is the relation
 * $$m_n(x_1,\ldots,x_n) = \langle x_1,\ldots, x_n\rangle + \Gamma$$,

where
 * $$\Gamma \in \sum_{j=1}^{n-1} \text{Im}(m_j)$$.

Hence all such $$A_\infty$$-enrichments on the cohomology algebra are related to one another.

Graded algebras from its ext algebra
Another structure theorem is the reconstruction of an algebra from its ext algebra. Given a connected graded algebra
 * $$A = k_A \oplus A_1 \oplus A_2 \oplus \cdots$$,

it is canonically an associative algebra. There is an associated algebra, called its Ext algebra, defined as
 * $$\operatorname{Ext}_A^\bullet(k_A,k_A)$$,

where multiplication is given by the Yoneda product. Then, there is an $$A_\infty$$-quasi-isomorphism between $$(A,0,m_2,0,\ldots)$$ and $$\operatorname{Ext}_A^\bullet(k_A,k_A)$$. This identification is important because it gives a way to show that all derived categories are derived affine, meaning they are isomorphic to the derived category of some algebra.