Abstract/Details

Fraïssé limits and colimits with applications to continua


2007 2007

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Abstract (summary)

The classical Fraïssé construction is a method of taking a direct limit of a family of finite models of a language provided the family fulfills certain amalgamation conditions. The limit is a countable model of the same language which can be characterized by its (injective) homogeneity and universality with respect to the initial family of models. A standard example is the family of finite linear orders for which the Fraïssé limit is the rational numbers with the usual ordering.

We present this classical construction via category theory, and within this context we introduce the dual construction. This respectively constitutes the Fraïssé colimits and limits indicated in the title. We provide several examples.

We then present the projective Fraïssé limit as a special case of the dual construction, and as such it is the categorical dual to the classical (injective) Fraïssé limit. In this dualization we use a notion of model theoretic structure which has a topological ingredient. This results in the countable limit structures being replaced by structures which are zero-dimensional, compact, second countable spaces with the property that the relations are closed and the functions are continuous.

We apply the theory of projective Fraïssé limits to the pseudo-arc by first representing the pseudo-arc as a natural quotient of a projective Fraïssé limit. Using this representation we derive topological properties of the pseudo-arc as consequences of the properties of projective Fraïssé limits. We thereby obtain a new proof of Mioduszewski's result that the pseudo-arc is surjectively universal among chainable continua, and also a homogeneity theorem for the pseudo-arc which is a strengthening of a result due to Lewis and Smith. We also find a new characterization of the pseudo-arc via the homogeneity property.

We continue with further applications of these methods to a class of continua known as pseudo-solenoids, and achieve analogous results for the universal pseudo-solenoid.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences, Continua, Fraisse limit, Pseudoarc, Pseudosolenoid
Title
Fraïssé limits and colimits with applications to continua
Author
Irwin, Trevor L.
Number of pages
97
Publication year
2007
Degree date
2007
School code
0093
Source
DAI-B 69/02, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9780549442165
Advisor
Solecki, Slawomir
Committee member
Bedford, Eric; Leivant, Daniel; Moss, Lawrence
University/institution
Indiana University
Department
Mathematics
University location
United States -- Indiana
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3297082
ProQuest document ID
304857268
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/304857268
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