Homotopy associative algebra

For a fixed field $k$ an $A\_\backslash infty$-algebra{{Citation |last=Lefèvre-Hasegawa |first=Kenji |title=Sur les A-infini catégories |date=2003-10-21 |arxiv=math/0310337 |id=arXiv:math/0310337}} is a $\backslash Z$-graded vector space

:$A\; =\; \backslash bigoplus\_\{p\; \backslash in\; \backslash Z\}A\_p$

equipped with morphisms $m\_i\; \backslash colon\; A^\{\backslash otimes\; i\}\; \backslash to\; A$ of degree $2\; -\; i$ for each $i\; \backslash geq\; 1$ satisfying a coherence condition: for all $n$,

:$\backslash sum\_\{j+k+l=n,\backslash ,j+1+l=i\}\; (-1)^\{jk+l\}\; m\_i(\backslash mathrm\{id\}^\{\backslash otimes\; j\}\; \backslash otimes\; m\_k\; \backslash otimes\; \backslash mathrm\{id\}^\{\backslash otimes\; l\})\; =\; 0$.

An $A\_\backslash infty$-morphism of $A\_\backslash infty$-algebras $f:\; A\; \backslash to\; B$ is a family of morphisms $f\_i:\; A^\{\backslash otimes\; i\}\; \backslash to\; B$ of degree $i-1$ satisfying a similar coherence condition: for all $n$,$$\backslash sum\_\{j+k+l=n,\backslash ,j+1+l=i\}\; (-1)^\{jk+l\}\; f\_i(\backslash mathrm\{id\}^\{\backslash otimes\; j\}\; \backslash otimes\; m\_k\; \backslash otimes\; \backslash mathrm\{id\}^\{\backslash otimes\; l\})\; =\; \backslash sum\_\{i\_1+\backslash cdots+i\_r=n\}\; (-1)^s\; m\_r(f\_\{i\_1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; f\_\{i\_r\})$$where $s\; =\; \backslash sum\_\{2\; \backslash leq\; u\; \backslash leq\; r\}((1-i\_u)\backslash sum\_\{i\; \backslash leq\; v\; \backslash leq\; u\}\; i\_v)$. (In both coherence conditions, the signs in the sums can be bypassed by shifting the grading by one.)

The coherence conditions are easy to write down for low degrees.

For $n\; =\; 1$ this is the condition that

:$m\_1(m\_1(a\_1))\; =\; 0$,

since $j+1+l=1$ gives $j=l=0$ and thus $k=1=i$. This means that $m\_1$ is a differential on $A$.

The coherence condition for $n=2$ gives $$m\_1\; \backslash circ\; m\_2\; (a\_1\; \backslash otimes\; a\_2)\; -\; m\_2\; \backslash circ\; (1\; \backslash otimes\; m\_1)(a\_1\; \backslash otimes\; a\_2)\; -\; m\_2\; \backslash circ\; (m\_1\; \backslash otimes\; 1)(a\_1\; \backslash otimes\; a\_2)\; =\; 0,$$or $$m\_1(m\_2(a\_1\; \backslash otimes\; a\_2))\; =\; m\_2(m\_1(a\_1)\; \backslash otimes\; a\_2)\; +\; (-1)^$$

In this degree the coherence condition reads

:$\backslash begin\{align\}$

m_2(m_2(a_1\otimes a_2)\otimes a_3) - m_2(a_1\otimes m_2(a_2\otimes a_3)) =& +m_3(m_1(a_1)\otimes a_2\otimes a_3) \\

&-m_3(a_1\otimes m_1(a_2)\otimes a_3) \\

& + m_3(a_1\otimes a_2\otimes m_1(a_3)) \\

& + m_1(m_3(a_1\otimes a_2 \otimes a_3)).

\end{align}

Notice that the left hand side of the equation is the failure of the multiplication $m\_2$ to make $A$ into an algebra which is associative on the nose. The right hand side is the differential on $A$ applied to the triple product plus the triple product applied to the differential on $A\; \backslash otimes\; A\; \backslash otimes\; A$, and says precisely that associativity holds up to a homotopy given by $m\_3$. In particular, we have that the multiplication induced by $m\_2$ on $H\_*(A,m\_1)$ is strictly associative.

Note if $m\_3=0$ then $(A,m\_1)$ is a differential graded algebra with multiplication $m\_2$, as the vanishing of $m\_3$ means that $m\_2$ is associative on the nose.

In higher degrees the coherency conditions give many different terms. We can arrange the right hand side to be a chain homotopy given by $m\_n$ as we did in the case of $n=3$:

:$\backslash begin\{align\}$

&\pm m_n(m_1(a_1) \otimes \cdots \otimes a_n)) \\ &\pm \cdots \\

&\pm m_n(a_1 \otimes \cdots \otimes m_1(a_n)) \\

&\pm m_1(m_n(a_1 \otimes \cdots \otimes a_n)),

\end{align}

while the terms on the left hand side indicate the failure of lower $m\_i$ terms to satisfy a kind of generalized associativity. In essence, this means that an $A\_\backslash infty$ algebra may fail to be "higher-associative" in every degree, but at every degree its failure to be so will be parametrized by a chain homotopy given by the higher multiplication in the next degree.

There is a nice diagrammatic formalism of algebras which is described in Algebra+Homotopy=Operad{{cite arXiv|last=Vallette|first=Bruno|date=2012-02-15|title=Algebra+Homotopy=Operad|class=math.AT|eprint=1202.3245}} 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.

Since the definition of an $A\_\backslash infty$-algebra requires an infinite sequence of higher multiplications, one might hope that there is a way to repackage the definition in terms of a single structure with finitely many operations. This is possible (after a little setup) by reinterpreting the $m\_i$ as components of a single map instead.

Given a (graded) vector space $V$, the reduced tensor coalgebra $\backslash overline\{T^c\}V$ on $V$ is $\backslash bigoplus\_\{n\; \backslash geq\; 1\}\; V^\{\backslash otimes\; n\}$ with the (non-cocommutative) coproduct $\backslash Delta\_\{\backslash overline\{T^c\}\}$ given by splitting of tensors, i.e., , where we write the internal tensor product of $\backslash overline\{T^c\}V$ with the standard tensor product symbol and the external tensor product used in defining a coproduct with the vertical stroke for clarity. Given any coalgebra $C$, there is a canonical filtration of $C$ defined by $F\_kC\; =\; \backslash ker\; \backslash Delta^\{(k+1)\}$, where $\backslash Delta^\{(2)\}\; =\; \backslash Delta\_C$, $\backslash Delta^\{(n)\}\; =\; (\backslash mathrm\{id\}^\{\backslash otimes\; (n-2)\}\; \backslash otimes\; \backslash Delta\_C)\; \backslash circ\; \backslash Delta^\{(n-1)\}$; $C$ is called cocomplete if $C\; =\; \backslash bigcup\_k\; F\_kC$. The reduced tensor coalgebra is the universal cocomplete coalgebra over $V$, i.e., for any other cocomplete coalgebra $C\text{'}$, there is a natural bijection between the coalgebra maps from $C\text{'}$ to $\backslash overline\{T^c\}V$ and the graded vector space maps $C\text{'}\; \backslash to\; V$.

A coderivation on a coalgebra $C$ is a $k$-module map $D:\; C\; \backslash to\; C$ satisfying the "co-Leibniz rule"$\backslash Delta\_C\; \backslash circ\; D\; =\; (D\; \backslash otimes\; \backslash mathrm\{id\}\; +\; \backslash mathrm\{id\}\; \backslash otimes\; D)\; \backslash circ\; \backslash Delta\_C$. The suspension $SV$ of a graded vector space $V$ is the graded vector space defined by $(SV)^i\; =\; V^\{i+1\}$.

With this notation, we have the following fact: an $A\_\backslash infty$-algebra structure on a graded vector space $A$ is the same thing as a coderivation $D$ on $\backslash overline\{T^c\}(SA)$ which is a differential, that is, $D^2\; =\; 0$. To see this, note that $D$ is determined by its composite with the quotient $\backslash overline\{T^c\}(SA)\; \backslash to\; SA$ since $\backslash overline\{T^c\}V$ is the cofree cocomplete coalgebra on $V$. We obtain the $m\_n$ by decomposing the composite into $b\_n:\; (SA)^\{\backslash otimes\; n\}\; \backslash to\; SA$ and unshifting the map to a map from $A^\{\backslash otimes\; n\}$ to $A$. The condition that $D$ be a coderivation yields $D\; =\; b\_1\text{'}\; +\; b\_2\text{'}\; +\; b\_3\text{'}\; +\; \backslash cdots$, with $b\_i\text{'}(v\_1\backslash otimes\; \backslash cdots\; \backslash otimes\; v\_j)\; =\; \backslash sum\_\{a+i+b=j\}\; v\_1\; \backslash otimes\; \backslash cdots\; \backslash otimes\; v\_a\; \backslash otimes\; b\_i(v\_\{a+1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; v\_\{a+i\})\; \backslash otimes\; v\_\{a+i+1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; v\_j$, and the condition that $D^2\; =\; 0$ yields $\backslash sum\_\{i+j-1=m\}\; b\_i\text{'}(b\_j\text{'}(v\_1\; \backslash otimes\; \backslash cdots\; \backslash otimes\; v\_m))\; =\; 0$ for all $m$, or equivalently $\backslash sum\_\{j+k+l=m,j+1+l=i\}\; b\_i(\backslash mathrm\{id\}^\{\backslash otimes\; j\}\; \backslash otimes\; b\_k\; \backslash otimes\; \backslash mathrm\{id\}^\{\backslash otimes\; l\})\; =\; 0$, which unshifts to the standard (signed) conditions on the $m\_n$ due to the sign rule for shifting complexes. The differential graded coalgebra $(\backslash overline\{T^c\}(SA),D)$ defined in this way is called the bar construction on $A$ and denoted $BA$.

Many notions are easier to write out by considering $A\_\backslash infty$-algebras via their bar constructions. For instance, a morphism $f:\; A\; \backslash to\; A\text{'}$ of $A\_\backslash infty$-algebras is equivalently a morphism of differential graded coalgebras $Bf:\; BA\; \backslash to\; BA\text{'}$, a quasiisomorphism of $A\_\backslash infty$-algebras is equivalently a quasiisomorphism of differential graded coalgebras, and a homotopy between $A\_\backslash infty$-algebra morphisms is equivalently a homotopy between differential graded coalgebra morphisms.

Every associative algebra $(A,\backslash cdot)$ has an $A\_\backslash infty$-infinity structure by defining $m\_2(a,b)\; =\; a\backslash cdot\; b$ and $m\_i\; =\; 0$ for $i\; \backslash neq\; 2$. Hence $A\_\backslash infty$-algebras generalize associative algebras.

Every differential graded algebra $(A^\backslash bullet,\; d)$ has a canonical structure as an $A\_\backslash infty$-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 $A\_\backslash infty$-structure on the graded cohomology algebra $HA^\backslash bullet$ 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

$\backslash text\{Ho\}(\backslash text\{dga\})\; \backslash simeq\; \backslash text\{Ho\}(A\_\backslash infty\backslash text\{-alg\})$

One of the motivating examples of $A\_\backslash infty$-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 $A\_\backslash infty$-algebra structure from its structure as an H-space.

Consider the graded algebra $V^\backslash bullet\; =\; V\_0\backslash oplus\; V\_1$ 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 $w$.{{Cite web|last1=Allocca|first1=Michael|last2=Lada|first2=Thomas|date=|title=A Finite Dimensional A-infinity algebra example|url=https://lada.math.ncsu.edu/FinDimAInfEx-final.pdf|url-status=live|archive-url=https://web.archive.org/web/20200928010418/https://lada.math.ncsu.edu/FinDimAInfEx-final.pdf|archive-date=28 Sep 2020|access-date=|website=}}{{Cite journal|last1=Daily|first1=Marilyn|last2=Lada|first2=Tom|date=2005|title=A finite dimensional $L_\infty$ algebra example in gauge theory|url=https://projecteuclid.org/euclid.hha/1139839375|journal=Homology, Homotopy and Applications|language=EN|volume=7|issue=2|pages=87–93|doi=10.4310/HHA.2005.v7.n2.a4|issn=1532-0073|doi-access=free}} Even in this simple example there is a non-trivial $A\_\backslash infty$-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 $(V^\backslash bullet)^\{\backslash otimes\; k\}$. Define the differential $m\_1$ by

:$\backslash begin\{align\}$

m_1(v_0) = w \\

m_1(v_1) = w,

\end{align}

and for $d\; \backslash geq\; 2$

:$\backslash begin\{align\}$

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

$\backslash begin\{align\}$

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

:$\backslash begin\{align\}$

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 $H^*(V^\backslash bullet,\; [m\_2])$ we have only the degree $0$ terms $v\_1,v\_2$ since $w$ is killed by the differential $m\_1$.

One of the key properties of $A\_\backslash infty$-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 $A\_\backslash infty$-algebra $A^\backslash bullet$ and a homotopy equivalence of complexes

:$f\backslash colon\; B^\backslash bullet\; \backslash to\; A^\backslash bullet$,

then there is an $A\_\backslash infty$-algebra structure on $B^\backslash bullet$ inherited from $A^\backslash bullet$ and $f$ can be extended to a morphism of $A\_\backslash infty$-algebras. There are multiple theorems of this flavor with different hypotheses on $B^\backslash bullet$ and $f$, some of which have stronger results, such as uniqueness up to homotopy for the structure on $B^\backslash bullet$ and strictness on the map $f$.{{cite arXiv|last=Burke|first=Jesse|date=2018-01-26|title=Transfer of A-infinity structures to projective resolutions|class=math.KT|eprint=1801.08933}}

One of the important structure theorems for $A\_\backslash infty$-algebras is the existence and uniqueness of minimal models – which are defined as $A\_\backslash infty$-algebras where the differential map $m\_1\; =\; 0$ is zero. Taking the cohomology algebra $HA^\backslash bullet$ of an $A\_\backslash infty$-algebra $A^\backslash bullet$ from the differential $m\_1$, so as a graded algebra,

:$HA^\backslash bullet\; =\; \backslash frac\{\backslash text\{ker\}(m\_1)\}\{m\_1(A^\backslash bullet)\}$,

with multiplication map $[m\_2]$. It turns out this graded algebra can then canonically be equipped with an $A\_\backslash infty$-structure,

:$(HA^\backslash bullet,\; 0,\; [m\_2],\; m\_3,m\_4,\backslash ldots)$,

which is unique up-to quasi-isomorphisms of $A\_\backslash infty$-algebras.{{cite arXiv|last=Kadeishvili|first=Tornike|date=2005-04-21|title=On the homology theory of fibre spaces|eprint=math/0504437}} In fact, the statement is even stronger: there is a canonical $A\_\backslash infty$-morphism

:$(HA^\backslash bullet,\; 0,\; [m\_2],\; m\_3,m\_4,\backslash ldots)\; \backslash to\; A^\backslash bullet$,

which lifts the identity map of $A^\backslash bullet$. 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 $H^*(D^b\_\backslash infty(X))$ 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 $\backslash mathcal\{Hom\}^\backslash bullet(\backslash mathcal\{F\}^\backslash bullet,\backslash mathcal\{G\}^\backslash bullet)$^{pg 586-593}. In this was, the degree $k$ morphisms in the category $H^*(D^b\_\backslash infty(X))$ are given by $\backslash text\{Ext\}(\backslash mathcal\{F\}^\backslash bullet,\backslash mathcal\{G\}^\backslash bullet)$.

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

Given a differential graded algebra $(A^\backslash bullet,\; d)$ its minimal model as an $A\_\backslash infty$-algebra $(HA^\backslash bullet,\; 0,\; [m\_2],\; m\_3,m\_4,\backslash ldots)$ is constructed using the Massey products. That is,

:$\backslash begin\{align\}$

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 $A\_\backslash infty$-algebra structure on $HA^\backslash bullet$ is closely related to this construction. Given another $A\_\backslash infty$-structure on $HA^\backslash bullet$ with maps $m\_i\text{'}$, there is the relation{{cite arXiv|last1=Buijs|first1=Urtzi|last2=Moreno-Fernández|first2=José Manuel|last3=Murillo|first3=Aniceto|date=2019-02-19|title=A-infinity structures and Massey products|class=math.AT|eprint=1801.03408}}

:$m\_n(x\_1,\backslash ldots,x\_n)\; =\; \backslash langle\; x\_1,\backslash ldots,\; x\_n\backslash rangle\; +\; \backslash Gamma$,

where

:$\backslash Gamma\; \backslash in\; \backslash sum\_\{j=1\}^\{n-1\}\; \backslash text\{Im\}(m\_j)$.

Hence all such $A\_\backslash infty$-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

:$A\; =\; k\_A\; \backslash oplus\; A\_1\; \backslash oplus\; A\_2\; \backslash oplus\; \backslash cdots$,

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

:$\backslash operatorname\{Ext\}\_A^\backslash bullet(k\_A,k\_A)$,

where multiplication is given by the Yoneda product. Then, there is an $A\_\backslash infty$-quasi-isomorphism between $(A,0,m\_2,0,\backslash ldots)$ and $\backslash operatorname\{Ext\}\_A^\backslash bullet(k\_A,k\_A)$. 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.

• A∞-category
• Associahedron
• Mirror symmetry conjecture
• Homological mirror symmetry
• Homotopy Lie algebra
• Derived algebraic geometry

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{{refend}}

Category:Homotopical algebra

Category:Homological algebra

Category:Algebraic geometry

Category:Homotopy theory