Continuity in the coarse and weak shape categories (Q1762382)

This work involves two types of shape theory which are different from the standard shape theory. These are, coarse shape theory [\textit{N. Koceić Bilan} and \textit{N. Uglešić}, Glas. Mat., III. Ser. 42, No. 1, 145--187 (2007; Zbl 1131.55005)] and weak shape theory [\textit{N. Uglešić} and \textit{B. Červar}, Int. J. Pure Appl. Math. 39, No. 3, 363--428 (2007; Zbl 1142.55008)]. The author deals with continuity theorems in these categories. Suppose that \(\mathcal{D}\subset\mathcal{C}\) is a pro-reflective subcategory of a category \(\mathcal{C}\), and \(Y\in Ob(\mathcal{C})\). Then every \(\mathcal{C}\)-expansion \(\mathbf q:Y\to\mathbf Y\) (a morphism of pro-\(\mathcal{C}\) with \(Y\) taken as a rudimentary inverse system) (with respect to \(\mathcal{D}\)) induces a limit \(S(\mathbf q):Y\to S(\mathbf Y)\) (morphism of pro-\(Sh_{(\mathcal{C},\mathcal{D})}\)) in the shape category \(Sh{(\mathcal{C},\mathcal{D})}\). Here, \(S:\mathcal{C} \to Sh{(\mathcal{C},\mathcal{D})}\) is the shape functor. This continuity theorem is proven for the coarse shape category \(Sh^*{(\mathcal{C},\mathcal{D})}\) as Theorem 1 and for the weak shape category \(Sh_*{(\mathcal{C},\mathcal{D})}\) as Theorem 2. As applications of the continuity theorems, three examples are given, one concerning the long segment \([0,\omega_1]\), a second about the ``long solenoid,'' and a third about a construction that leads to a Hausdorff continuum. The author seems only to have speculative conclusions about the latter example.