Lemma 1

The relation

“$\alpha$ is the class of a $\tau$-extension for Iso$_{\tau}$”

is collectivizing in $\alpha$.

Set $E_0= F\times G$, and consider the mappings $\iota_0: F\rightarrow E_0$ and $\pi_0: E_0\rightarrow G$ defined by $\iota_0(f) = (f, e)$ for $f\in F$ and $\pi_0(f, g) =g$ for $(f, g)\in F\times G$. Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$-extension of G by F. The mapping $\pi$ is surjective, so has a section $\sigma : G\rightarrow \Gamma$ such that $\sigma (e) =e$. The mapping $u: (f, g)\mapsto \iota (f)\sigma (g)$ from $F\times G$ to Γ is bijective. We endow $F\times G$ with the group law obtained by transfer of structure. The triple $(E_0, \iota_0, \pi_0)$ is then a $\tau$-extension isomorphic to $\mathscr{E}$. Lemma 1 now follows from Set Theory, II, §6, No. 6, p. 122.

We denote by Ex$_{\tau}(G,F)$ the set of classes of $\tau$-extensions for the relation Iso$_{\tau}$.