3 a clarification

Let me mention Sullivan's minimal models.

Every commutative differential graded $\mathbb{Q}$-algebra (cdga) $A^*$ concentrated in non-negative degrees and such that $H^0(A^*)=\mathbb{Q}$ admits a minimal Sullivan model $i:M^*\to A^*$ where $M^*$ is a free commutative graded algebra obtained from $\mathbb{Q}$ by adding generators of non-negative degrees so that the differential of each generator is a $\mathbb{Q}$-linear combination of products of length $\geq 2$ of the previous generators, and $i$ is a map of cgda's that induces a cohomology isomorphism (i.e., a quasi-isomorphism).

The minimal model is unique up to a non-unique isomorphism. More generally, if $f:A^*\to B^*$ is a map of cdga's and $j:N^*\to B^*$ is a minimal model of $B^*$, then there is a cdga map $g:M^*\to N^*$, defined up to cdga homotopy, such that $fi=gj$ up to cdga homotopy; moreover, if $f$ is a quasi-isomorphism, then $g$ is an isomorphism.

This reduces the classification of non-negative cdga's up to quasi-isomorphism (and as a consequence, the classification of simply connected topological spaces up to rational homotopy) to the classification of algebras of a certain kind up to isomorphism.

Of course, this example is similar to some mentioned before (in a sense it is the commutative analog of the answer of John Palmieri).

2 a clarification

Let me mention Sullivan's minimal models.

Every commutative differential graded $\mathbb{Q}$-algebra (cdga) $A^*$ concentrated in non-negative degrees and such that $H^0(A^*)=\mathbb{Q}$ admits a minimal Sullivan model $i:M^*\to A^*$ where $M^*$ is a free commutative graded algebra obtained from $\mathbb{Q}$ by adding generators of non-negative degrees so that the differential of each generator is a sum $\mathbb{Q}$-linear combination of products of length $\geq 2$ of the previous generators, and $i$ is a map of cgda's that induces a cohomology isomorphism (i.e., a quasi-isomorphism).

The minimal model is unique up to a non-unique isomorphism. More generally, if $f:A^*\to B^*$ is a map of cdga's and $j:N^*\to B^*$ is a minimal model of $B^*$, then there is a cdga map $g:M^*\to N^*$, defined up to homotopy, such that $fi=gj$ up to homotopy; moreover, if $f$ is a quasi-isomorphism, then $g$ is an isomorphism.

This reduces the classification of non-negative cdga's up to quasi-isomorphism (and as a consequence, the classification of simply connected topological spaces up to rational homotopy) to the classification of algebras of a certain kind up to isomorphism.

Of course, this example is similar to some mentioned before (in a sense it is the commutative analog of the answer of John Palmieri).

1 [made Community Wiki]

Let me mention Sullivan's minimal models.

Every commutative differential graded $\mathbb{Q}$-algebra (cdga) $A^*$ concentrated in non-negative degrees and such that $H^0(A^*)=\mathbb{Q}$ admits a minimal Sullivan model $i:M^*\to A^*$ where $M^*$ is a free commutative graded algebra obtained from $\mathbb{Q}$ by adding generators of non-negative degrees so that the differential of each generator is a sum of products of length $\geq 2$ of the previous generators, and $i$ is a map of cgda's that induces a cohomology isomorphism (i.e., a quasi-isomorphism).

The minimal model is unique up to a non-unique isomorphism. More generally, if $f:A^*\to B^*$ is a map of cdga's and $j:N^*\to B^*$ is a minimal model of $B^*$, then there is a map $g:M^*\to N^*$, defined up to homotopy, such that $fi=gj$ up to homotopy; moreover, if $f$ is a quasi-isomorphism, then $g$ is an isomorphism.

This reduces the classification of non-negative cdga's up to quasi-isomorphism (and as a consequence, the classification of simply connected topological spaces up to rational homotopy) to the classification of algebras of a certain kind up to isomorphism.

Of course, this example is similar to some mentioned before (in a sense it is the commutative analog of the answer of John Palmieri).