Affiliation:
1. Faculty of Mathematics “Al.I.Cuza” University , 700505 - Iaşi , Romania
Abstract
Abstract
This paper has two parts. In the first part we present a case in which we can calculate in a simple way the product of two inverse systems as object in a pro-category of a category with directed products, without counting systems at the same set of indices. In the second part, which is related to some papers of Keesling [4], Kodama [5], Mardešić [7], and Dydak-Mardešić [10], we give a sufficient condition by which a directed product of a category 𝒯 is transformed by a shape functor S : 𝒯 → Sh(𝒯, 𝒫) into a directed product. This theorem is applied to the toplogical shape category Sh(Top) where some examples are obtained.
Reference11 articles.
1. [1] W. Holsztyński, An extension and axiomatic charcaterization of Borsuk’s theory of shape, Fund. Math., 70 (1971), 1105-1108.10.4064/fm-70-2-157-168
2. [2] I.C. Isaksen, A model structure for the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001), 2805-2841.10.1090/S0002-9947-01-02722-2
3. [3] I.C. Isaksen, Calculating limits and colimits in pro-categories, Fundam.Math. 175, No.2, 175-194 (2002).
4. [4] J. Keesling, Products in the shape category and some applications, Symp. Math. Inst. Nazionale di Alta Matematica 16 (1973), Academic Press, New York, 1974, 133-142.
5. [5] Y. Kodama, On shape of product spaces, General Topology and its Applications 8 (1978) 141-150.10.1016/0016-660X(78)90045-4