What's the meaning of "the set of isomorphic classes of extensions of $\mathcal F$ by $\mathcal G$"?

Question

I think an isomorphic class of extensions of $\mathcal F$ by $\mathcal G$ is a proper class, but a proper class cann't be an element of a set, so what's the meaning of "the set of isomorphic classes of extensions of $\mathcal F$ by $\mathcal G$"?

Answer 1

You are right, the extensions of $\mathcal F$ by $\mathcal G$ do not form a set, but a proper class $\mathfrak E$. Thus it is not correct to speak about the set of isomorphism classes. However, such phrases are frequently used. It should be interpreted in the following sense:

There exists a set $\mathbf E$ of extensions of $\mathcal F$ by $\mathcal G$ such that each extension of $\mathcal F$ by $\mathcal G$ is isomorphic to some element of $\mathbf E$.

Then you can form the set of isomorphism classes in $\mathbf E$. In the best case you can achieve that the set $\mathbf E$ has the property that isomorphism classes are singletons. That is, $\mathbf E$ may be regarded as a set of representatives of isomorphims classes of $\mathfrak E$.

In the case of your question this point is covered by the Proposition. To each element of the $Ext^1$-set one assigns an extension, thereby producing the set $\mathbf E$.

However, we must be aware that constructions like that in your question ("equivalence classes in a proper class") do not automatically admit a set of representatives. This has to be proved in each case.