A note on delay-inverse systems, I
Nikica Uglešić
TL;DR
This work investigates a generalized delay-inverse framework that extends inverse systems via a delay mechanism and defines the delay-pro-category. The central result shows that, for indexing sets with cardinality $|A|$ among $\{\aleph_{0}, \aleph_{1}, \dots, \aleph_{n+1}\}$, any delay-inverse system is isomorphic in $Dpro$-$\mathcal{C}$ to a cofinite, commutative inverse system, effectively reducing to the classical pro-category. The analysis uses the Mardešić trick and transfinite induction to construct commutative representatives and proves that isomorphism types in $Dpro$-$\mathcal{C}$ coincide with those in $pro$-$\mathcal{C}$ for these cardinalities. The results indicate that while delay-inverse techniques provide a conceptual tool for studying complex objects, they do not yield new invariants in the standard cardinal regimes, though they may still assist in coarse shape analyses.
Abstract
A generalization of an inverse system in a category was recently introduced, as well as that of the corresponding pro-category These so called the delay-inverse systems and delay-pro-category could potentially yield a new theory of (delay-) inverse systems as well as a kind of coarser abstract shape theory. However, we have proven that, whenever an indexing set has cardinality $\aleph_{n}, n\in\mathbb{N}_{0}$, the potential new theory reduces, in its essence (the classification and invariants), to the ordinary one.