A DG category is an $A_\infty$-category with $m_n = 0$ for $n \geq 3$. In this case:

• $m_1 = d$ (the differential).
• $m_2 =$ composition (strictly associative).
• $m_3 = m_4 = \cdots = 0$.

The $A_\infty$-relation for $n = 3$ reduces to $m_2(m_2(a, b), c) = m_2(a, m_2(b, c))$, i.e., strict associativity. Thus DG categories are the "strict" case of $A_\infty$-categories.

An $A_\infty$-algebra is an $A_\infty$-category with a single object. It consists of a graded vector space $A$ with operations $m_n: A^{\otimes n} \to A$ of degree $2 - n$ satisfying the $A_\infty$-relations.

The classical example: for a topological space $X$, the singular cochain complex $C^*(X; k)$ is a DG algebra under the cup product, but it also carries a natural $A_\infty$-structure that encodes the higher Massey products.

An $A_\infty$-category (or algebra) is minimal if $m_1 = 0$ (the differential vanishes). The homological perturbation lemma shows that every $A_\infty$-category is quasi-isomorphic to a minimal one.

For a minimal $A_\infty$-algebra $H^*(\mathcal{A})$, the operations $m_n$ for $n \geq 3$ encode the "higher Massey product" information that is lost when passing to cohomology. The minimal model is unique up to $A_\infty$-isomorphism.

The Fukaya category $\operatorname{Fuk}(M, \omega)$ of a symplectic manifold $(M, \omega)$ is an $A_\infty$-category with:

• Objects: Lagrangian submanifolds $L$ (with extra data: grading, spin structure, flat line bundle).
• $\operatorname{Hom}(L_0, L_1)$: the Floer cochain complex $CF^*(L_0, L_1)$, generated by intersection points.
• $m_1$: counts pseudo-holomorphic strips (Floer differential).
• $m_2$: counts pseudo-holomorphic triangles (pair-of-pants).
• $m_n$: counts pseudo-holomorphic $(n+1)$-gons.

The $A_\infty$-relations follow from the compactification of the moduli space of pseudo-holomorphic polygons (Gromov compactness + gluing). Composition is not strictly associative because the moduli space of quadrilaterals has a boundary consisting of two types of degenerate configurations.

For modules $M, N$ over a ring $R$, the Ext algebra $\operatorname{Ext}^*_R(M, M)$ has a natural $A_\infty$-structure (its minimal model). The operation $m_2$ is the Yoneda product, and the higher operations $m_n$ encode Massey products.

This $A_\infty$-structure determines the derived category $D^b(\operatorname{mod}\text{-}R)$ near $M$: the $A_\infty$-modules over $\operatorname{Ext}^*_R(M, M)$ recover the subcategory generated by $M$.

Given a DG category $\mathcal{A}$ and a deformation retract of its Hom complexes onto their cohomology, the transfer theorem produces a minimal $A_\infty$-structure on $H^*(\mathcal{A})$ such that the inclusion $H^*(\mathcal{A}) \hookrightarrow \mathcal{A}$ extends to an $A_\infty$-quasi-isomorphism.

The transferred operations $m_n$ are computed by summing over planar rooted trees with $n$ leaves, where each internal vertex contributes a homotopy and each edge contributes a projection or inclusion. This is the homological perturbation lemma (HPL) in the $A_\infty$ setting.

The Maurer--Cartan equation in an $A_\infty$-algebra $A$ is:

$\sum_{n=1}^{\infty} m_n(a, \ldots, a) = 0 \quad \text{for } a \in A^1$

Solutions parametrize deformations of the underlying object. The deformation theory of an associative algebra $B$ is controlled by the $A_\infty$-structure on its Hochschild cochain complex $C^*(B, B)$.

An $A_\infty$-functor $F: \mathcal{A} \to \mathcal{B}$ between $A_\infty$-categories consists of:

• A map on objects $F: \operatorname{Ob}(\mathcal{A}) \to \operatorname{Ob}(\mathcal{B})$.
• For each $n \geq 1$, multilinear maps $F_n: \operatorname{Hom}(X_{n-1}, X_n) \otimes \cdots \otimes \operatorname{Hom}(X_0, X_1) \to \operatorname{Hom}(F(X_0), F(X_n))$ of degree $1 - n$.

These satisfy compatibility equations with the $A_\infty$-structures. In particular, $F_1$ need not be a chain map; the failure is corrected by $F_2$, and so on. When $F_n = 0$ for $n \geq 2$, we recover a DG functor (or strict $A_\infty$-functor).

An $A_\infty$-functor $F: \mathcal{A} \to \mathcal{B}$ is a quasi-isomorphism (or quasi-equivalence) if:

1. Each $F_1: \operatorname{Hom}_{\mathcal{A}}(X, Y) \to \operatorname{Hom}_{\mathcal{B}}(F(X), F(Y))$ induces an isomorphism on cohomology.
2. The induced functor $H^0(F): H^0(\mathcal{A}) \to H^0(\mathcal{B})$ is essentially surjective.

A fundamental theorem (Kadeishvili, Kontsevich--Soibelman) states that every $A_\infty$-quasi-isomorphism has an $A_\infty$-quasi-inverse: an $A_\infty$-functor $G: \mathcal{B} \to \mathcal{A}$ with $G \circ F$ and $F \circ G$ homotopic to the identity.

An $A_\infty$-natural transformation (or pre-natural transformation) $\eta: F \Rightarrow G$ between $A_\infty$-functors $F, G: \mathcal{A} \to \mathcal{B}$ consists of multilinear maps:

$\eta_n: \operatorname{Hom}(X_{n-1}, X_n) \otimes \cdots \otimes \operatorname{Hom}(X_0, X_1) \to \operatorname{Hom}(F(X_0), G(X_n))$

for $n \geq 0$, satisfying compatibility equations. When $n = 0$, $\eta_0$ provides a "component" $\eta_0(X) \in \operatorname{Hom}(F(X), G(X))$ for each object $X$, and the higher components encode the homotopy coherence of naturality.

Every $A_\infty$-category is quasi-isomorphic to a DG category. The construction proceeds by taking the "bar construction": for an $A_\infty$-category $\mathcal{A}$, the category of $A_\infty$-modules $\operatorname{Mod}_{\mathcal{A}}^{A_\infty}$ is naturally a DG category (after passing to the bar complex), and $\mathcal{A}$ embeds into it via the Yoneda functor.

The advantage of $A_\infty$-categories is that they admit minimal models ($m_1 = 0$), which do not generally exist in the DG world (since a DG category with $d = 0$ has trivial homotopy theory).

An $A_\infty$-category over $k$ gives rise to a $k$-linear $\infty$-category via the DG nerve construction (applied to the equivalent DG category). The passage is:

$A_\infty\text{-Cat}_k \to \mathbf{dgCat}_k \xrightarrow{N_{\mathrm{dg}}} \operatorname{Cat}_\infty^{k\text{-linear}}$

In the other direction, any $k$-linear stable $\infty$-category with a set of generators can be modeled by an $A_\infty$-category (its full subcategory on the generators).

An $A_\infty$-algebra is an algebra over the $A_\infty$-operad, which is a cofibrant resolution of the associative operad $\operatorname{Assoc}$ in the model category of operads. Similarly:

• $E_\infty$-algebras are algebras over a cofibrant resolution of $\operatorname{Comm}$ (commutative operad).
• $L_\infty$-algebras are algebras over a cofibrant resolution of $\operatorname{Lie}$ (Lie operad).

The $A_\infty$-operad is the minimal resolution of $\operatorname{Assoc}$ and is governed by the Stasheff associahedra $K_n$ (convex polytopes whose faces encode the $A_\infty$-relations).