Vanjski i unutarnji pristup konačnom grubom obliku

doctoral thesis

• Title (english): External and intrinsic approach to the finite coarse shape
• Author: Ivan Jelić
• Mentor: Nikola Koceić Bilan (mentor)
• Committee member: Goran Erceg (predsjednik povjerenstva)
• Committee member: Nikola Koceić Bilan (član povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Vlasta Matijević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-08, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U radu definiramo novu oblikovnu kategoriju $Sh^{\circledast}$ koju ćemo nazvati kategorijom konačnoga gruboga oblika. Ta će kategorija imati iste objekte kao kategorije oblika i gruboga oblika, ali će morfizmi među tim objektima biti drugačiji. Kategoriju konačnoga gruboga oblika konstruiramo korištenjem teorije inverznih sustava i poliedarskih ekspanzija topoloških prostora, to jest, vanjskim pristupom. Definirat ćemo dva odgovarajuća vjerna funktora, jednoga iz kategorije oblika u... More kategoriju konačnoga gruboga oblika, a drugoga iz kategorije konačnoga gruboga oblika u kategoriju gruboga oblika. Primjerima ćemo pokazati da ti funktori nisu puni, to jest, da je kategorija konačnoga gruboga oblika prava natkategorija kategorije oblika i prava potkategorija kategorije gruboga oblika. Kategoriju konačnoga gruboga oblika za kompaktne metričke prostore opisat ćemo i \textit{unutarnjim pristupom}. Da bismo to postigli, restringirat ćemo klasu objekata na skup svih zatvorenih podskupova Hilbertove kocke $Q$, a teoriju inverznih sustava zamijeniti teorijom $\epsilon$-neprekidnih funkcija. Stoga ćemo prvo detaljno razraditi teoriju $\epsilon$-neprekidnosti definirajući osnovne pojmove i dokazujući najvažnija svojstva $\epsilon$-neprekidnih funkcija i relacije $\epsilon$-homotopije. Potom generaliziramo Borsukove fundamentalne i aproksimativne nizove te Sanjurjove približavajuće nizove definirajući $\circledast$-fundamentalne, $\circledast$-aproksimativne te $\circledast$-približavajuće nizove redom. Na skupovima $\circledast$-fundamentalnih, $\circledast$-aproksimativnih i $\circledast$-približavajućih nizova definirat ćemo odgovarajuće relacije ekvivalencije čije će klase biti morfizmi novih kategorija $Sh^{\circledast}_f$, $Sh^{\circledast}_a$ i $InSh^{\circledast}$ redom. Kategoriju $InSh^{\circledast}$ nazvat ćemo \textit{kategorijom unutarnjega konačnoga gruboga oblika}. Nadalje, definiramo tri odgovarajuća funktora, jedan među kategorijama $Sh^{\circledast}|_Q$ (restrikcija kategorije $Sh^{\circledast}$ na zatvorene podskupove od $Q$) i $Sh^{\circledast}_f$, jedan među kategorijama $Sh^{\circledast}_f$ i $Sh^{\circledast}_a$ te jedan među kategorijama $Sh^{\circledast}_a$ i $InSh^{\circledast}$. Dokazat ćemo da će tako definirani funktori biti kategorijski izomorfizmi, što će značiti da je $InSh^{\circledast}$ unutarnja reinterpretacija kategorije konačnoga gruboga oblika zatvorenih podskupova Hilbertove kocke. Uz to, definirat ćemo vjeran funktor među kategorijama $InSh$ i $InSh^{\circledast}$ koji objekte drži fiksnima, a svakom morfizmu unutarnjega oblika pridružuje inducirani morfizam unutarnjega konačnoga gruboga oblika. Stoga ćemo postojeću Sanjurjovu kategoriju $InSh$ unutarnjega oblika smatrati pravom potkategorijom nove kategorije $InSh^{\circledast}$ unutarnjega konačnoga gruboga oblika. Konačno, dokazat ćemo da unutarnji konačni grubi oblik ne ovisi o ulaganju metričkoga prostora u Hilbertovu kocku $Q$, odnosno, da su svaka dva smještenja proizvoljnoga kompaktnoga metričkoga prostora istoga unutarnjega konačnoga gruboga oblika. Time ćemo klasifikaciju po unutarnjem konačnom grubom obliku proširiti na cijelu klasu $\mathcal{MC}pt$ svih kompaktnih metričkih prostora i pritom dokazati da su kompaktni metrički prostori $M$ i $M'$ istoga konačnoga gruboga oblika ako i samo ako su $M$ i $M'$ istoga unutarnjega konačnoga gruboga oblika. Less

Abstract (english)

In this thesis we will define a new shape category $Sh^{\circledast}$ of topological spaces called the finite coarse shape category. Category $Sh^{\circledast}$ will have the same class of objects as the categories of shape and coarse shape, but morphisms will be different. We will contruct the finite coarse shape category by using polyhedral expansions and inverse systems theory, i.e., using an external approach. Two appropriate faithful functors will be defined, one of them... More from the shape category $Sh$ to the finite coarse shape category $Sh^{\circledast}$ and the other one from the finite coarse shape category $Sh^{\circledast}$ to the coarse shape category $Sh^*$. We will prove by the examples that those functors are not full, i.e., that the finite coarse shape category is a proper subcategory of the coarse shape category and that it contains the shape category as its proper subcategory. The coarse shape category of metric compacta will also be given an intrinsic approach. To achieve that, we will restrict the class of objects to the set of all closed subsets of the Hilbert cube $Q$ and replace the inverse systems theory by the theory of $\epsilon$-continuous functions. Therefore we shall give a detailed overview of the $\epsilon$-continuity theory by defining the main notions and proving the most important properties of the $\epsilon$-continuous functions and $\epsilon$-homotopy. We will generalise Borsuk's fundamental and approximative sequences and Sanjurjo's proximate sequences by defining so called $\circledast$-fundamental, $\circledast$-approximative and $\circledast$-proximate sequences, respectively. On the sets of the $\circledast$-fundamental, $\circledast$-approximative and $\circledast$-proximate sequences between any two closed subsets of the Hilbert cube $Q$ we will define an appropriate equivalence relations, classes of which will be morphisms of the new categories $Sh^{\circledast}_f$, $Sh^{\circledast}_a$ and $InSh^{\circledast}$, respectively. Category $InSh^{\circledast}$ will be called the intrinsic finite coarse shape category}. Futhermore, three new functors will be defined, one of them between the categories $Sh^{\circledast}|_Q$ (the restriction of the $Sh^{\circledast}$ category to the set of all closed subsets of $Q$) and $Sh^{\circledast}_f$, the second one between $Sh^{\circledast}_f$ and $Sh^{\circledast}_a$ and the third one between the categories $Sh^{\circledast}_a$ and $InSh^{\circledast}$. We will prove that each of these three functors is an isomorphism, which means that category $InSh^{\circledast}$ is an intrinsic reinterpretation of the finite coarse shape category $Sh^{\circledast}|_Q$ of all closed subsets of $Q$. We will also define a faithful functor from $InSh$ to $InSh^{\circledast}$ which is an identity on the set of objects and which associates each intrinsic shape morphism with the induced intrinsic finite coarse shape morphism. Hence, Sanjurjo's intrinsic shape category $InSh$ may be considered as a proper subcategory of the constructed intrinsic finite coarse shape category $InSh^{\circledast}$. In the last section we will prove that the intrinsic finite coarse shape does not depend on the embedding of a compact metric space into the Hilbert cube $Q$, i.e., that every two embeddings of any compact metric space have the same intrinsic finite coarse shape. That will allow us to extend the classification by the intrinsic finite coarse shape to the whole class $\mathcal{MC}pt$ of metric compacta. Finally, we will prove that two compact metric spaces $M$ and $M'$ have the same finite coarse shape if and only if $M$ and $M'$ have the same intrinsic finite coarse shape. Less

• Keywords: oblik; homotopija; kategorija; klasifikacija; topologija
• Keywords (english): shape; homotopy; category; classification; topology
• Language: croatian
• URN:NBN: urn:nbn:hr:217:404652
• Promotion: 2022
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: vii, 137 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-01-24 13:37:37