The finite coarse shape - inverse systems approach and intrinsic approach

Given an arbitrary category \(\mathcal{C}\), a category \(pro^{*^f}\)-\(\mathcal{C}\) is constructed such that the known \(pro\)-\(\mathcal{C}\) category may be considered as a subcategory of \(pro^{*^f}\)-\(\mathcal{C}\) and that \(pro^{*^f}\)-\(\mathcal{C}\) may be considered as a subcategory of \(pro^*\)-\(\mathcal{C}\). Analogously to the construction of the shape category \(Sh_{(\mathcal{C},\mathcal{D})}\) and the coarse category \(Sh^*_{(\mathcal{C},\mathcal{D})}\), an (abstract) finite coarse shape category \(Sh^{*^f}_{(\mathcal{C},\mathcal{D})}\) is obtained. Between these three categories appropriate faithful functors are defined. The finite coarse shape is also defined by an intrinsic approach using the notion of the \(\epsilon\)-continuity. The isomorphism of the finite coarse shape categories obtained by these two approaches is constructed. Besides, an overview of some basic properties related to the notion of the \(\epsilon\)-continuity is given.