We had the adjoint pair

with

The adjunction induces a bijection on the homotopy categories of the subcategories:

Here, cofibrant can just be though of as can be replaced by Sullivan algebras, and finite type refers to where and each is finite-dimensional. Cofibrant:

Two maps of CDGAs are homotopic if

where .

If is a Sullivan algebra, then being homotopic is an equivalence relation.

preserves homotopies, that is

Consider the diagram

Let such that Then

for every Sullivan algebra (or more generally cofibrant algebra).

Consider the diagram

Note that are weak equivalences (as is contractible). This leads to the following diagram:

Therefore

is a quasi-isomorphism (reason: singular cohomology turns weak-equivalences into isomorphisms). Now, is a Sullivan algebra, therefore

We have .

Therefore

The adjunction

induces a bijection

It is enough to prove that the bijection of the adjunction

and its inverse preserve homotopies.

We have

where is the unit of the adjunction. The following are true:

Therefore preserves homotopies.

Recall that

where is the generator in degree . Then

is simply connected because is by definition . Recall that

One can check that .