Algebraic Structures up to Homotopy

Roughly, by an algebraic structure I mean a space equipped with a bunch of structure maps which are subject to certain relations. In general, algebraic structures are rigid and do not behave well with respect to homotopy operations on their underlying space. However, some algebraic structures are sufficiently flexible and much better behaved in this regard. We call these algebraic structure up to homotopy. Sounds vague and confusing? In this post, we’ll consider a concrete and easy example: associative algebras.

Associative algebras up to homotopy

We will work in the differential graded (or dg for short) category, meaning that for us a space is a dg vector space or chain complex. Let $(A,\mathrm{d}^A,m)$ be a dg associative algebra, i.e. an associative algebra in chain complexes. Explicitly, this means that $(A,\mathrm{d}^A)$ is a chain complex, the multiplication $m\colon A\otimes A \to A$ is a chain map $\partial(m) = 0,$ and $m$ satisfies the associativity condition $$ m\circ(m\otimes 1) - m\circ(1\otimes m) = 0 . $$ For any two multilinear maps $f\colon V^{\otimes n} \to V$ and $g\colon V^{\otimes k} \to V,$ we denote by $f\circ_i g\colon V^{\otimes(n+k-1)} \to V$ their $i$-th partial composition $$ f \circ_i g = f \circ \big(1^{\otimes(i-1)}\otimes g\otimes 1^{\otimes(n-i)}\big) . $$ Using this notation, the associativity condition can be written as $m \circ_1 m - m \circ_2 m = 0.$

Now consider the following situation: let $(V,\mathrm{d}^V)$ be a deformation retract of the chain complex $(A,\mathrm{d}^A),$ i.e. $$ h \circlearrowright (A,\mathrm{d}^A) \underset{i}{\overset{p}{\rightleftarrows}} (V,\mathrm{d}^V), \quad \text{such that } \begin{cases} \mathrm{id}_A - i\circ p = [\mathrm{d}^A,h] , \cr \mathrm{id}_V - p\circ i = 0 . \end{cases} $$ Using these data, we can define a multiplication map on $V$ by the composition $$ \mu_2\colon V\otimes V \xrightarrow{i\otimes i} A\otimes A \xrightarrow{m} A \xrightarrow{p} V . $$ Of course, when $h=0,$ $i$ and $p$ are inverse isomorphisms and it is easy to see that in this case $\mu_2$ defines a dg associative algebra structure on $(V,\mathrm{d}^V).$ However, in general, this multiplication $\mu_2$ is not associative. In fact, a simple computation shows that \begin{align} \hspace{2em}&\hspace{-2em}\mu_2\circ_1\mu_2 - \mu_2\circ_2\mu_2 \cr &= p \circ (m \circ_1 m - m \circ_2 m) \circ (i,i,i) \cr &\quad- \partial\big( p \circ (m \circ_1 hm - m \circ_2 hm) \circ (i,i,i) \big) , \end{align} of which the first term vanishes by associativity of $m.$ Thus, if we define $\mu_3 = p \circ (m \circ_1 hm - m \circ_2 hm) \circ (i,i,i),$ then $$ \mu_2\circ_1\mu_2 - \mu_2\circ_2\mu_2 = - \partial(\mu_3) , $$ i.e. $\partial(\mu_3)$ measures the defect of associativity of $\mu_2.$

Continuing along the same lines, one defines $\mu_n$ for all $n\geq 2$ in a similar fashion and as a result obtains an infinite sequence of higher coherence conditions for the associativity relation. The structure $(V,\mathrm{d}^V,{\mu_n})$ defined in this way is known as an associative algebra up to homotopy or $\mathrm{A}_\infty$-algebra. This notion was introduced by Stasheff [8].

Definition. An $\mathrm{A}_\infty$-algebra $(A,\mathrm{d},\mu_n)$ consists of a chain complex $(A,\mathrm{d})$ with structure maps \begin{align} &\mu_n\colon A^{\otimes n} \to A[n-2] , &&n \geq 2. \end{align} These structure maps are required to satisfy the following homotopy associativity conditions: $$ \partial(\mu_n) + \sum_{\llap{i+j}=\rlap{n+1}} \ (-1)^{(j-1)i} \cdot \sum_{p=1}^j \ (-1)^{(p-1)(i-1)} \cdot \mu_j \circ_p \mu_i = 0. $$

Morphisms of homotopy associative algebras

An obvious notion of morphism exists for $\mathrm{A}_\infty$-algebras: a chain map $f\colon (V,\textrm{d}) \to (V’,\textrm{d}^\prime)$ commuting with all structure maps, i.e. $f\circ\mu_n = \mu’_n\circ(f,\dotsc,f).$ Such a morphism is often called a strict morphism of $A_\infty$-algebras.

But let’s step back and have another look at the situation above: assume $(A,\textrm{d}^A,m)$ is a dg associative algebra, $(V,\textrm{d}^V)$ a deformation retract of $(A,\textrm{d}^A),$ and $(V,\textrm{d}^V,\mu_n)$ is the $\mathrm{A}_\infty$-algebra as defined above. Consider the chain map $i.$ Of course, we would like it to be a morphism of $\mathrm{A}_\infty$-algebras. However, a simple computation using our definition of $\mu_2$ shows that \begin{align} i\circ\mu_2 - m\circ(i,i) &= i\circ p\circ m\circ(i,i) - m\circ(i,i) \cr &= \partial( - h \circ m \circ (i,i) ) . \end{align} Clearly, the definition of a strict morphism is too restrictive for our purpose. The computation suggests that a good definition of morphism should contain higher homotopy terms measuring the defect of commutativity with the structure maps. In our case, we define $i_2 = - h \circ m \circ (i,i)$ and obtain $$ i\circ\mu_2 - m\circ(i,i) = \partial( i_2 ) . $$

As for the algebras, one can iterate this construction and thereby obtain the more suitable definition for morphisms of $\mathrm{A}_\infty$-algebras stated below.

Definition. Let $(A,\textrm{d},\mu_n)$ and $(A’,\textrm{d}^\prime,\mu’_n)$ be $\mathrm{A}_\infty$-algebras. A homotopy morphism or $\infty$-morphism of $\mathrm{A}_\infty$-algebras $f\colon (A,\textrm{d},\mu_n) \to (A’,\textrm{d}^\prime,\mu’_n)$ consists of components \begin{align} &f_n\colon A^{\otimes n} \to A’[n-1] , &&n \geq 1. \end{align} These components are required to satisfy the following coherence conditions: \begin{align} \partial(f_n) &- \sum_{\llap{i+j}=\rlap{n+1}} \ (-1)^{(j-1)i} \cdot \sum_{p=1}^j \ (-1)^{(p-1)(i-1)} \cdot f_j \circ_p \mu_i \cr &+ \sum_{\substack{1\leq j\leq n\cr \llap{i_1}+\dotsb+\rlap{i_j=n}}} \varepsilon \cdot \mu’_j \circ (f_{i_1},\dotsc,f_{i_j}) = 0, \end{align} where the missing sign is given by $$ \varepsilon = (-1)^{(j-1)(n-j)} \cdot (-1)^{\sum_{p=1}^j(p-1)(i_p-1)} . $$

The Homotopy Transfer Theorem

We have defined $\mathrm{A}_\infty$-algebras and their $\infty$-morphisms in such a way that a certain homotopy transfer result holds for deformation retracts. In fact, the result holds even when $(A,\textrm{d}^A,\mu_n)$ is any $\mathrm{A}_\infty$-algebra instead of just a dg associative algebra.

Theorem (HTT [3]). Let $(A,\textrm{d}^A,\mu_n)$ be an $\mathrm{A}_\infty$-algebra and $(V,\textrm{d}^V)$ a deformation retract of $(A,\textrm{d}^A).$ Then $(V,\textrm{d}^V)$ can be equipped with structure maps $\mu’_n$ in such a way that $(V,\textrm{d}^V,\mu’_n)$ becomes an $\mathrm{A}_\infty$-algebra and $i$ extends to a homotopy morphism of $\mathrm{A}_\infty$-algebras.

Note that under some conditions, in particular when working over a field, any (co)chain complex contains its (co)homology complex as a deformation retract. This gives a particularly interesting special case of the above theorem which we will explore below.

Hochschild cohomology and Massey products

In 1945, Hochschild introduced a cohomology theory for associative algebras [2] which is now known as Hochschild cohomology. We state the definition here as a reminder.

Given an associative algebra $(A,m),$ its Hochschild cochain complex is defined as $\textrm{HC}^p(A,A)=\textrm{Hom}(A^{\otimes p},A)$ with codifferential given by \begin{align} \hspace{2em}&\hspace{-2em}(\delta f)(x_1,\dotsc,x_{p+1}) \cr &= m(x_1,f(x_2,\dotsc,x_{p+1})) \cr &\quad+ \sum_{i=1}^p (-1)^i \cdot f(x_1,\dotsc,m(x_i,x_{i+1}),\dotsc,x_{p+1}) \cr &\quad+ (-1)^{p+1} \cdot m(f(x_1,\dotsc,x_p),x_{p+1}) . \end{align} Gerstenhaber [1] investigated the algebraic structure of the Hochschild cohomology complex. He introduced the cup product in Hochschild cohomology, which we recall below, and showed that it turns the cohomology complex into what is now known as a Gerstenhaber algebra.

To define the cup product in cohomology, we must first define it on the level of cochains, $$ \smile\colon \textrm{HC}^p(A,A) \otimes \textrm{HC}^q(A,A) \to \textrm{HC}^{p+q}(A,A) . $$ Here it is given by $$ (f\smile g)(x_1,\dotsc,x_{p+q}) = m(f(x_1,\dotsc,x_p),g(x_{p+1},\dotsc,x_{p+q})) . $$ It is obvious from the definition that the cup product on cochains is associative. In addition, one can easily check that $\smile$ is compatible with the codifferential $\delta$ and hence, for any associative algebra $(A,m),$ the Hochschild cochain complex $(\textrm{HC}^\bullet(A,A),\delta,\smile)$ is a dg associative algebra.

Using the HTT of the previous section, we can transfer the dg associative algebra structure of $(\textrm{HC}^\bullet(A,A),\delta,\smile)$ to its cohomology and, as a result, obtain an $A_\infty$-algebra $(\textrm{HH}^\bullet(A,A),0,\mu_n)$ with trivial differential $\textrm{d}=0$ and structure maps $\mu_n.$ These structure maps $$ \mu_n\colon (\textrm{HH}^\bullet(A,A))^{\otimes n} \to (\textrm{HH}^\bullet(A,A))[2-n] $$ generalize the classical Massey products as defined in [7]. For more details, we refer the reader to [6, §9.4.10].

Other types of homotopy algebras

Similar notions of algebra up to homotopy exist for other types of algebras, e.g. for Lie algebras [4,5]. For a modern textbook introduction to the subject, see [6].


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