You can define $Ext^n$ as the set of isomorphism classes of $n$-step extensions, equipped with the Baer sum. This eliminates choices from the definition of $Ext$ in exactly the same spirit as your original post eliminates choices from the definition of the trace.

You can define $\mathrm{Ext}^n$ as the set of isomorphism classes of $n$-step extensions, equipped with the Baer sum. This eliminates choices from the definition of $\mathrm{Ext}$ in exactly the same spirit as your original post eliminates choices from the definition of the trace.

You can define $Ext^n$ as the set of isomorphism classes of $n$-step extensions, equipped with the Baer sum. This eliminates choices from the definition of $Ext$ in exactly the same spirit as your original post eliminates chocies from the definition of the trace.

You can define $Ext^n$ as the set of isomorphism classes of $n$-step extensions, equipped with the Baer sum. This eliminates choices from the definition of $Ext$ in exactly the same spirit as your original post eliminates choices from the definition of the trace.