Any DG algebra $A$ over a field has a minimal model, which is a **minimal $A_\infty$-algebra** $(H,m_3,m_4,\dots)$. It consists of a graded algebra $B=H^*(A)$ equipped with multi-linear operations of degree $2-n$:
$$m_n\colon B\otimes\stackrel{n}{\cdots}\otimes B\longrightarrow B,\qquad n\geq 3,$$
satisfying some equations. One can reconstruct the DGA $A$ out of the minimal model, up to quasi-isomorphism.

**Triple Massey products** are a well-known secondary cohomology operation in the cohomology $B=H^*(A)$ of a DG algebra $A$. They are related to Toda brackets in the derived category $D^c(A)$ of compact $A$-modules. These Toda brackets are just defined in terms of the triangulated category structure of $D^c(A)$, i.e. in terms of the exact triangles. The ternary operation $m_3\colon B\otimes B\otimes B\rightarrow B$ computes triple Massey products.

There’s the truncated notion of **minimal $A_n$-algebra** $(B,m_3,\dots,m_n)$. There’s also an **obstruction theory** to increase $n$. Classical obstructions live in Hochschild cohomology $HH^{\bullet,*}(B)$. This is not surprising because the $m_i$ are Hochschild cochains. If all obstructions vanish one can extend the minimal $A_n$-algebra to a minimal $A_\infty$-algebra, and then construct a DG algebra $A$ out of it. There are even obstructions to the uniqueness of $A$, up to quasi-isomorphism.

If $\mathcal{T}$ is an essentially small algebraic triangulated category over a field and $X\in\mathcal{T}$ is a generator, one wonders about how many essentially different DG algebras $A$ exist such that $D^c(A)\simeq \mathcal{T}$ in such a way that $A\mapsto X$. This would imply that $B=H^*(A)=\mathcal{T}^*(X,X)$, the graded endomorphism algebra of $X$ in $\mathcal{T}$. Moreover, the ternary operation $m_3$ of a minimal model $(\mathcal{T}^*(X,X),m_3,m_4,\dots)$ of $A$ should compute Toda brackets in $\mathcal{T}$.

If $X$ is an **additive generator** of $\mathcal{T}$, the operation $m_3$ is completely determined by the triangulated structure of $\mathcal{T}$, so one is interested in the obstruction theory to extend $(\mathcal{T}^*(X,X),m_3)$ to an $A_\infty$-algebra. Classical obstructions don’t vanish, but there’s an **enhanced obstruction theory** living in the pages of a spectral sequence with $E_2=HH(\mathcal{T}^*(X,X))$ (almost everywhere).

Fernando Muro, “Enhanced A∞-Obstruction Theory,” Journal of Homotopy and Related Structures 15, no. 1 (March 1, 2020): 61–112, https://doi.org/10.1007/s40062-019-00245-0.

The differential $d_2$ is highly non-trivial but $E_3$ is concentrated in a narrow band, so narrow that $d_3=0$ so the spectral sequence stops at $E_3$ and we deduce that $A$ is essentially unique.

Fernando Muro, “Enhanced Finite Triangulated Categories,” Journal of the Institute of Mathematics of Jussieu 21, no. 3 (May 2022): 741–83, https://doi.org/10.1017/S1474748020000250.

This extends to the case where $X$ is **$d\mathbb{Z}$-cluster tilting**. In this case, the spectral sequence is so sparse that $E_2=\cdots=E_{d+1}$ on the nose, $d_{d+1}$ is highly non-trivial, but $E_{d+2}$ is concentrated in a narrow band and the spectral sequence stops here.

Gustavo Jasso, Bernhard Keller, and Fernando Muro, “The Derived Auslander-Iyama Correspondence,” August 24, 2023, https://doi.org/10.48550/arXiv.2208.14413.