arXiv subjects

Nikica Uglešić

Publications and source records attributed to Nikica Uglešić.

3 recordsLinked to original sources

arXiv2026

Basic Theory of Tower Systems

For a Banach space $X$, we study the completed direct limit of its even iterated duals along the canonical embeddings and the bounded inverse limit of the corresponding odd duals. We give explicit constructions, establish their duality relation, and describe the comparison maps associated with iterated dualization. We then examine several assertions in three preprints of Uglešić, giving counterexamples concerning dimension stability and the commutation of bidualization with direct limits. The finite quotient shape (the category whose objects are the finite-dimensional quotients, with all continuous linear maps between them) is classified by the algebraic dimension of the continuous dual. We also compute the dimensions of the finite iterated duals and the completed tower of $\ell^{\infty}(\mathbb{N})$, and prove that, inside the tower space, the canonical copy of a higher layer need not be the range of an $M$-projection. Throughout, uncompleted limits, Banach completions, and the different meanings of isomorphism are distinguished explicitly.

math.FA↗

arXiv2025

A note on delay-inverse systems, I

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.

math.CT↗

arXiv2019

Enriched pro-categories and shapes

Given a category $\mathcal C$ and a directed partially ordered set $J$, a certain category $pro^J -\mathcal C$ on inverse systems in $\mathcal C$ is constructed such that the ordinary pro-category $pro-\mathcal C$ is the most special case of a singleton $J \equiv \{1\}$. Further, the known $pro^*$-category $pro ^*-\mathcal C$ becomes $pro ^{\mathbb N }-\mathcal C$. Moreover, given a pro-reflective category pair $(\mathcal C, \mathcal D)$, the $J$-shape category $Sh^J_{(C,\mathcal D)}$ and the corresponding $J$-shape functor $S^J$ are constructed which, in mentioned special cases, become the well known ones. Among several important properties, the continuity theorem for a J-shape category is established. It implies the "$J$-shape theory" is a genuine one such that the shape and the coarse shape theory are its very special examples.

math.CT↗