General categorical framework for topologically free normed modules (Q2018018)

Following \textit{A. Y. Helemskiǐ} [Sb. Math. 204, No. 7, 1056--1083 (2013); translation from Mat. Sb. 204, No. 7, 127--158 (2013; Zbl 1288.46032)], a \textit{rigging} of a category \textbf{K} is a faithful functor \(\square\) from \textbf{K} to another category \textbf{L}. A \textit{rigged category} is a category \textbf{K} endowed with a rigging \(\square:\mathbf{K}\rightarrow\text\textbf{L}\). A \textbf{K}-morphism \(\varphi\) is an \textit{admissible epimorphism} provided that \(\square\varphi\) is an \textbf{L}-retraction. A \textbf{K}-object \(P\) is \textit{projective} provided that for every \textbf{K}-morphisms \(\varphi:P\rightarrow X\leftarrow Y:\sigma\) with an admissible \(\sigma\), there exists a \textbf{K}-morphism \(\psi:P\rightarrow Y\) such that \(\sigma\circ\psi=\varphi\). In Section~3, the author describes projective objects in the category of normed modules, which is equipped with a rigging to the categories of bornological, normed, and linear spaces [loc. cit.]. Section 4 starts with the already known result that if \((\mathbf{K},\square)\) is a rigged category such that \(\square\) has a left adjoint, then for every \textbf{K}-object \(X\), there exists a \(\square\)-free \textbf{K}-object \(F\) and an admissible epimorphism \(\tau:F\rightarrow X\). The author generalizes this result, to include the category of normed modules with a rigging to the category of bornological spaces (Proposition 10 on page 49 and Example 7 on pages 49--50).