Glasnik Matematicki, Vol. 57, No. 1 (2022), 89-117. \( \)

THE FINITE COARSE SHAPE - INVERSE SYSTEMS APPROACH AND INTRINSIC APPROACH

Ivan Jelić and Nikola Koceić Bilan

Faculty of Science, University of Split, 21 000 Split, Croatia
e-mail:ivajel@pmfst.hr

Faculty of Science, University of Split, 21 000 Split, Croatia
e-mail:koceic@pmfst.hr

Abstract.   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.

2020 Mathematics Subject Classification.   55P55, 54C56

Key words and phrases.   Topological space, metric space, ANR, category, homotopy, shape, \(\epsilon\)-continuity

https://doi.org/10.3336/gm.57.1.07

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