Injective and epi-projective objects in categories of convex spaces (Q1273640)

A convex space, a notion introduced by \textit{D. Pumplün} [``Banach spaces and superconvex modules'', Symp. Gaussiana, Conf. A, Walter de Gruyter, Berlin, 323-338 (1995; Zbl 0855.46010)], \textit{D. Pumplün} and \textit{H. Röhrl} [``Convexity theories. IV: Klein-Hilbert parts in convex modules'', Appl. Categ. Struct. 3, No. 2, 173-200 (1995; Zbl 0823.52004)], is a set \(M\) with formal convex combinations, i.e., for any \(\alpha_i\geq 0\), \(1\leq i\leq n\), with \(\sum^n_{i=1}\alpha_i=1\), and any \(x_i\in M\), \(1\leq i\leq n\), a formal sum \(\sum^n_{i=1}\alpha_ix_i\in M\) exists, and these formal sums obey the usual computational rules of convex combinations. An affine mapping \(f:M_1\to M_2\) between convex spaces is a mapping preserving these formal convex combinations. The convex spaces with the affine mappings constitute the category \({\mathcal C}onv\) of convex spaces. Typical examples of convex spaces are convex subsets of real linear spaces. If \(C\subset E\) is a bounded, closed, convex subset of a complete Hausdorff topological real linear space \(E\), \(C\) is even closed under countably convex combinations, i.e., for \(c_i\in C\), \(\alpha_i\geq 0\), \(i\in\mathbb{N}\) and \(\sum^\infty_{i=1}\alpha_i=1\), \(\sum^\infty_{i=1}\alpha_ic_i\) converges and lies in \(C\) [cp. \textit{G. J. O. Jameson}, ``Convex series''. Proc. Camb. Philos. Soc. 72, 37-47 (1972; Zbl 0235.46019)]. This leads to the definition of superconvex spaces. A superconvex space \(M\) is a set closed under formal countably convex combinations, i.e., for any \(\alpha_i\geq 0\), \(\sum^\infty_{i=1}\alpha_i=1\), and \(x_i\in M\), \(i\in\mathbb{N}\), a formal sum \(\sum^\infty_{i=1}\alpha_ix_i\in \)M exists, and these formal sums obey the usual rules of convex combinations. Together with mappings preserving these formal sums the superconvex spaces form the category \({\mathcal S}{\mathcal C}\) of superconvex spaces. The author characterizes the injective objects and the \({\mathcal E}\)-projective objects, for different types of classes \({\mathcal E}\) of epimorphisms, in the categories \({\mathcal C}onv\), \({\mathcal S}{\mathcal C}\), the categories of (finitely) positively convex, of absolutely convex and of totally convex spaces.