Generalised algebraic models


First Edition


Algebraic theories and algebraic categories offer an innovative and revelatory description of the syntax and the semantics. An algebraic theory is a concrete mathematical object -- the concept -- namely a set of variables together with formal symbols and equalities between these terms; stated otherwise, an algebraic theory is a small category with finite products. An algebra or model of the theory is a set-theoretical interpretation -- a possible meaning -- or, more categorically, a finite product-preserving functor from the theory into the category of sets. We call the category of models of an algebraic theory an algebraic category. By generalising the theory we do generalise the models. This concept is the fascinating aspect of the subject and the reference point of our project. We are interested in the study of categories of models. We pursue our task by considering models of different theories and by investigating the corresponding categories of models they constitute. We analyse localizations (namely, fully faithful right adjoint functors whose left adjoint preserves finite limits) of algebraic categories and localizations of presheaf categories. These are still categories of models of the corresponding theory.We provide a classification of localizations and a classification of geometric morphisms (namely, functors together with a finite limit-preserving left adjoint), in both the presheaf and the algebraic context.


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Specifications


Publisher
Presses universitaires de Louvain
Title Part
Numéro 2
Author
Claudia Centazzo,
Collection
Thèses de la Faculté des sciences
Language
English
Publisher Category
Natural Sciences > Mathematics > Algebra, Geometry and Logic
Publisher Category
Natural Sciences > Mathematics
BISAC Subject Heading
SCI000000 SCIENCE
Onix Audience Codes
06 Professional and scholarly
CLIL (Version 2013-2019)
3051 SCIENCES FONDAMENTALES
Title First Published
01 January 2004
Type of Work
Thesis

Paperback


Publication Date
01 January 2004
ISBN-13
9782930344782
Extent
Main content page count : 194
Code
70957
Dimensions
16 x 24 x 17 cm
Weight
291 grams
List Price
16.40 €
ONIX XML
Version 2.1, Version 3

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Contents


Introduction 1

1 The theory of locally D-presentable categories 9

1.1 D-filtered categories . . . . . . . . . . . . . . . . . . . . . 9

- D-flat functors and sound doctrines . . . . . . . . . . . . 12

1.2 D-accessible categories . . . . . . . . . . . . . . . . . . . . 15

- The D-ind completion . . . . . . . . . . . . . . . . . . . 15

- The full subcategory of D-presentable objects . . . . . . 18

1.3 Locally D-presentable categories . . . . . . . . . . . . . . 22

- Restricting Yoneda . . . . . . . . . . . . . . . . . . . . . 25

1.4 D-completeness and Cauchy completeness . . . . . . . . . 26

2 The duality theorem for a limit doctrine 29

2.1 The duality Theorem . . . . . . . . . . . . . . . . . . . . . 29

- The 2-functor " . . . . . . . . . . . . . . . . . . . . . . . 29

- The objects of D-cont[A, set] . . . . . . . . . . . . . . . 32

- The 2-functor " is a biequivalence . . . . . . . . . . . . . 35

2.2 Some applications . . . . . . . . . . . . . . . . . . . . . . 39

- Comparing sound doctrines . . . . . . . . . . . . . . . . 39

2.3 A characterisation in terms of strong generators . . . . . . 54

- The Dop-colimit completion . . . . . . . . . . . . . . . . 55

- Strong generators play an important role . . . . . . . . . 56

- Consequences . . . . . . . . . . . . . . . . . . . . . . . . 62

3 On geometric morphisms and localizations of presheaf categories 69

3.1 Rambling about the concept of localization . . . . . . . . 69

- Exact and/or extensive categories . . . . . . . . . . . . . 70

- The exact completion of Fam . . . . . . . . . . . . . . . 75

- Some conditions our functors might satisfy . . . . . . . . 83

3.2 The localizations of presheaf categories . . . . . . . . . . . 87

- Comparing conditions . . . . . . . . . . . . . . . . . . . 94

- A sheaf-theoretical proof . . . . . . . . . . . . . . . . . . 98

4 On geometric morphisms and localizations of algebraic categories 121

4.1 Algebraic theories and algebraic categories . . . . . . . . . 121

4.2 A miscellany of preliminary results . . . . . . . . . . . . . 125

4.3 Some more conditions our functors might satisfy . . . . . 127

4.4 The coproduct completion and the exact completion in the case of an algebraic theory . .. 131

4.5 The localizations of (one sorted) algebraic categories . . . 152

4.6 The link with other characterisation theorems for localizations of algebraic categories . . 166

Acknowledgements 177

Bibliography 181