Homotopy associative algebra

For a fixed field ${\displaystyle k}$ an ${\displaystyle A_{\infty }}$-algebra[1] is a ${\displaystyle \mathbb {Z} }$-graded vector space

${\displaystyle A=\bigoplus _{p\in \mathbb {Z} }A^{p}}$

such that for ${\displaystyle d\geq 1}$ there exist degree ${\displaystyle 2-d}$, ${\displaystyle k}$-linear maps

${\displaystyle m_{d}\colon (A^{\bullet })^{\otimes d}\to A^{\bullet }}$

which satisfy a coherence condition:

${\displaystyle \sum _{\begin{matrix}1\leq p\leq d\\0\leq q\leq d-p\end{matrix}}(-1)^{\alpha }m_{d-p+1}(a_{d},\ldots ,a_{p+q+1},m_{p}(a_{p+q},\ldots ,a_{q+1}),a_{q},\ldots ,a_{1})=0}$,

where ${\displaystyle \alpha =(-1)^{{\text{deg}}(a_{1})+\cdots +\deg(a_{q})-q}}$.

The coherence conditions are easy to write down for low degrees[1]^{pgs 583–584}.

For ${\displaystyle d=1}$ this is the condition that

${\displaystyle m_{1}(m_{1}(a_{1}))=0}$,

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

Unpacking the coherence condition for ${\displaystyle d=2}$ gives the degree ${\displaystyle 0}$ map ${\displaystyle m_{2}}$. In the sum there are the inequalities

${\displaystyle {\begin{matrix}1\leq p\leq 2\\0\leq q\leq 2-p\end{matrix}}}$

of indices giving ${\displaystyle (p,q)}$ equal to ${\displaystyle (1,0),(1,1),(2,0)}$. Unpacking the coherence sum gives the relation

${\displaystyle m_{2}(a_{2},m_{1}(a_{1}))+(-1)^{\deg(a_{1})-1}m_{2}(m_{1}(a_{2}),a_{1})+m_{1}(m_{2}(a_{1},a_{2}))=0}$,

which when rewritten with

${\displaystyle (-1)^{\deg a}m_{1}(a)=d(a)}$ and ${\displaystyle (-1)^{\deg a_{1}}m_{2}(a_{2},a_{1})=a_{2}\cdot a_{1}}$

as the differential and multiplication, it is

${\displaystyle d(a_{2}\cdot a_{1})=(-1)^{\deg(a_{1})}d(a_{2})\cdot a_{1}+a_{2}\cdot d(a_{1})}$,

which is the Leibniz rule for differential graded algebras.

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

${\displaystyle {\begin{aligned}m_{2}(m_{2}(a\otimes b)\otimes c)-m_{2}(a\otimes m_{2}(b\otimes c))=&\pm m_{3}(m_{1}(a)\otimes b\otimes c)\\&\pm m_{3}(a\otimes m_{1}(b)\otimes c)\\&\pm m_{3}(a\otimes b\otimes m_{1}(c))\\&\pm m_{1}(m_{3}(a\otimes b\otimes c)).\end{aligned}}}$

Notice that the left hand side of the equation is the failure for ${\displaystyle m_{2}}$ to be an associative algebra on the nose. One of the inputs for the first three ${\displaystyle m_{3}}$ maps are coboundaries since ${\displaystyle m_{1}}$ is the differential, so on the cohomology algebra ${\displaystyle (H^{*}(A^{\bullet },m_{1}),[m_{2}])}$ these elements would all vanish since ${\displaystyle m_{1}(a)=m_{1}(b)=m_{1}(c)=0}$. This includes the final term ${\displaystyle m_{1}(m_{3}(a\otimes b\otimes c))}$ since it is also a coboundary, giving a zero element in the cohomology algebra. From these relations we can interpret the ${\displaystyle m_{3}}$ map as a failure for the associativity of ${\displaystyle m_{2}}$, meaning it is associative only up to homotopy.

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

${\displaystyle {\begin{aligned}&\pm m_{d}(a_{d},\ldots ,a_{2},m_{1}(a_{1}))\\&\pm \cdots \\&\pm m_{d}(m_{1}(a_{d}),a_{d-1},\ldots ,a_{1})\\&\pm m_{1}(m_{d}(a_{d},\ldots ,a_{1})).\end{aligned}}}$

In degree ${\displaystyle d=4}$ the other terms can be written out as

${\displaystyle {\begin{aligned}&\pm m_{3}(m_{2}(a_{4},a_{3}),a_{2},a_{1})\\&\pm m_{3}(a_{4},m_{2}(a_{3},a_{2}),a_{1})\\&\pm m_{3}(a_{4},a_{3},m_{2}(a_{2},a_{1}))\\&\pm m_{2}(m_{3}(a_{4},a_{3},a_{2}),a_{1})\\&\pm m_{2}(a_{4},m_{3}(a_{3},a_{2},a_{1})),\end{aligned}}}$

showing how elements in the image of ${\displaystyle m_{3}}$ and ${\displaystyle m_{2}}$ interact. This means the homotopy of elements, including one that's in the image of ${\displaystyle m_{2}}$ minus the multiplication of elements where one is a homotopy input, differ by a boundary. For higher order ${\displaystyle d>4}$, these middle terms can be seen how the middle maps ${\displaystyle m_{2},\ldots ,m_{d-1}}$ behave with respect to terms coming from the image of another higher homotopy map.

There is a nice diagrammatic formalism of algebras which is described in Algebra+Homotopy=Operad[4] 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.

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

Every differential graded algebra ${\displaystyle (A^{\bullet },d)}$ has a canonical structure as an ${\displaystyle A_{\infty }}$-algebra[1] where ${\displaystyle m_{1}=d}$ and ${\displaystyle m_{2}}$ is the multiplication map. All other higher maps ${\displaystyle m_{i}}$ are equal to ${\displaystyle 0}$. Using the structure theorem for minimal models, there is a canonical ${\displaystyle A_{\infty }}$-structure on the graded cohomology algebra ${\displaystyle HA^{\bullet }}$ 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

${\displaystyle {\text{Ho}}({\text{dga}})\simeq {\text{Ho}}(A_{\infty }{\text{-alg}})}$

of differential graded algebras and ${\displaystyle A_{\infty }}$-algebras.

One of the motivating examples of ${\displaystyle A_{\infty }}$-algebras comes from the study of H-spaces. Whenever a topological space ${\displaystyle X}$ is an H-space, its associated singular chain complex ${\displaystyle C_{*}(X)}$ has a canonical ${\displaystyle A_{\infty }}$-algebra structure from its structure as an H-space.[3]

Consider the graded algebra ${\displaystyle V^{\bullet }=V_{0}\oplus V_{1}}$ over a field ${\displaystyle k}$ of characteristic ${\displaystyle 0}$ where ${\displaystyle V_{0}}$ is spanned by the degree ${\displaystyle 0}$ vectors ${\displaystyle v_{1},v_{2}}$ and ${\displaystyle V_{1}}$ is spanned by the degree ${\displaystyle 1}$ vector ${\displaystyle w}$.[5][6] Even in this simple example there is a non-trivial ${\displaystyle A_{\infty }}$-structure which gives differentials in all possible degrees. This is partially due to the fact there is a degree ${\displaystyle 1}$ vector, giving a degree ${\displaystyle k}$ vector space of rank ${\displaystyle 1}$ in ${\displaystyle (V^{\bullet })^{\otimes k}}$. Define the differential ${\displaystyle m_{1}}$ by

${\displaystyle {\begin{aligned}m_{1}(v_{0})=w\\m_{1}(v_{1})=w,\end{aligned}}}$

and for ${\displaystyle d\geq 2}$

${\displaystyle {\begin{aligned}m_{d}(v_{1}\otimes w^{\otimes k}\otimes v_{1}\otimes w^{\otimes (d-2)-k})&=(-1)^{k}s_{d}v_{1}&0\leq k\leq d-2\\m_{d}(v_{1}\otimes w^{\otimes (d-2)}\otimes v_{2})&=s_{d+1}v_{1}\\m_{d}(v_{1}\otimes w^{\otimes (d-1)})&=s_{d+1}w,\end{aligned}}}$

where ${\displaystyle m_{n}=0}$ on any map not listed above and ${\displaystyle s_{n}=(-1)^{(n-1)(n-2)/2}}$. In degree ${\displaystyle d=2}$, so for the multiplication map, we have ${\displaystyle {\begin{aligned}m_{2}(v_{1},v_{1})&=-v_{1}\\m_{2}(v_{1},v_{2})&=v_{1}\\m_{2}(v_{1},w)&=w.\end{aligned}}}$ And in ${\displaystyle d=3}$ the above relations give

${\displaystyle {\begin{aligned}m_{3}(v_{1},v_{1},w)&=v_{1}\\m_{3}(v_{1},w,v_{1})&=-v_{1}\\m_{3}(v_{1},w,v_{2})&=-v_{1}\\m_{3}(v_{1},w,w)&=-w.\end{aligned}}}$

When relating these equations to the failure for associativity, there exist non-zero terms. For example, the coherence conditions for ${\displaystyle v_{1},v_{2},w}$ will give a non-trivial example where associativity doesn't hold on the nose. Note that in the cohomology algebra ${\displaystyle H^{*}(V^{\bullet },[m_{2}])}$ we have only the degree ${\displaystyle 0}$ terms ${\displaystyle v_{1},v_{2}}$ since ${\displaystyle w}$ is killed by the differential ${\displaystyle m_{1}}$.

One of the key properties of ${\displaystyle A_{\infty }}$-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 ${\displaystyle A_{\infty }}$-algebra ${\displaystyle A^{\bullet }}$ and a homotopy equivalence of complexes

${\displaystyle f\colon B^{\bullet }\to A^{\bullet }}$,

then there is an ${\displaystyle A_{\infty }}$-algebra structure on ${\displaystyle B^{\bullet }}$ inherited from ${\displaystyle A^{\bullet }}$ and ${\displaystyle f}$ can be extended to a morphism of ${\displaystyle A_{\infty }}$-algebras. There are multiple theorems of this flavor with different hypotheses on ${\displaystyle B^{\bullet }}$ and ${\displaystyle f}$, some of which have stronger results, such as uniqueness up to homotopy for the structure on ${\displaystyle B^{\bullet }}$ and strictness on the map ${\displaystyle f}$.[7]

One of the important structure theorems for ${\displaystyle A_{\infty }}$-algebras is the existence and uniqueness of minimal models – which are defined as ${\displaystyle A_{\infty }}$-algebras where the differential map ${\displaystyle m_{1}=0}$ is zero. Taking the cohomology algebra ${\displaystyle HA^{\bullet }}$ of an ${\displaystyle A_{\infty }}$-algebra ${\displaystyle A^{\bullet }}$ from the differential ${\displaystyle m_{1}}$, so as a graded algebra,

${\displaystyle HA^{\bullet }={\frac {{\text{ker}}(m_{1})}{m_{1}(A^{\bullet })}}}$,

with multiplication map ${\displaystyle [m_{2}]}$. It turns out this graded algebra can then canonically be equipped with an ${\displaystyle A_{\infty }}$-structure,

${\displaystyle (HA^{\bullet },0,[m_{2}],m_{3},m_{4},\ldots )}$,

which is unique up-to quasi-isomorphisms of ${\displaystyle A_{\infty }}$-algebras.[8] In fact, the statement is even stronger: there is a canonical ${\displaystyle A_{\infty }}$-morphism

${\displaystyle (HA^{\bullet },0,[m_{2}],m_{3},m_{4},\ldots )\to A^{\bullet }}$,

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

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 ${\displaystyle H^{*}(D_{\infty }^{b}(X))}$ of the dg-category consisting of cochain complexes of coherent sheaves on a non-singular variety ${\displaystyle X}$ over a field ${\displaystyle k}$ of characteristic ${\displaystyle 0}$ and morphisms given by the total complex of the Cech bi-complex of the differential graded sheaf ${\displaystyle {\mathcal {Hom}}^{\bullet }({\mathcal {F}}^{\bullet },{\mathcal {G}}^{\bullet })}$[1]^{pg 586-593}. In this was, the degree ${\displaystyle k}$ morphisms in the category ${\displaystyle H^{*}(D_{\infty }^{b}(X))}$ are given by ${\displaystyle {\text{Ext}}({\mathcal {F}}^{\bullet },{\mathcal {G}}^{\bullet })}$.

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

Given a differential graded algebra ${\displaystyle (A^{\bullet },d)}$ its minimal model as an ${\displaystyle A_{\infty }}$-algebra ${\displaystyle (HA^{\bullet },0,[m_{2}],m_{3},m_{4},\ldots )}$ is constructed using the Massey products. That is,

${\displaystyle {\begin{aligned}m_{3}(x_{3},x_{2},x_{1})&=\langle x_{3},x_{2},x_{1}\rangle \\m_{4}(x_{4},x_{3},x_{2},x_{1})&=\langle x_{4},x_{3},x_{2},x_{1}\rangle \\&\cdots &\end{aligned}}}$

It turns out that any ${\displaystyle A_{\infty }}$-algebra structure on ${\displaystyle HA^{\bullet }}$ is closely related to this construction. Given another ${\displaystyle A_{\infty }}$-structure on ${\displaystyle HA^{\bullet }}$ with maps ${\displaystyle m_{i}'}$, there is the relation[9]

${\displaystyle m_{n}(x_{1},\ldots ,x_{n})=\langle x_{1},\ldots ,x_{n}\rangle +\Gamma }$,

where

${\displaystyle \Gamma \in \sum _{j=1}^{n-1}{\text{Im}}(m_{j})}$.

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

Another structure theorem is the reconstruction of an algebra from its ext algebra. Given a connected graded algebra

${\displaystyle A=k_{A}\oplus A_{1}\oplus A_{2}\oplus \cdots }$,

it is canonically an associative algebra. There is an associated algebra, called its Ext algebra, defined as

${\displaystyle \operatorname {Ext} _{A}^{\bullet }(k_{A},k_{A})}$,

where multiplication is given by the Yoneda product. Then, there is an ${\displaystyle A_{\infty }}$-quasi-isomorphism between ${\displaystyle (A,0,m_{2},0,\ldots )}$ and ${\displaystyle \operatorname {Ext} _{A}^{\bullet }(k_{A},k_{A})}$. This identification is important because it gives a way to show that all derived categories are derived affine, meaning they are isomorphic the derived category of some algebra.