Equivalence of coherent theories (Q675238)

\textit{Yu. T. Lisitsa} and \textit{S. Mardešić} [Glas. Mat., III. Ser. 19(39), 335-399 (1984; Zbl 0553.55009)] developed notions of coherent maps and coherent homotopies between commutative inverse systems. In this paper, the author uses a category of coherent inverse systems as introduced by \textit{Yu. T. Lisitsa} [Sov. Math., Dokl. 25, 373-378 (1982); translation from Dokl. Akad. Nauk SSSR 263, 532-536 (1982; Zbl 0518.55009)]. In this paper, the question as to whether or not the Lisitsa-Mardešić form yields a full subcategory of this larger category is addressed. It is shown how to take a coherent map between commutative inverse systems and to deform it to a special coherent map, i.e. a coherent map in the sense of Lisitsa and Mardešić. This technical result clarifies considerably the detailed relationship between these very similar approaches to strong shape theory.