Presses universitaires de Louvain
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20241113
eng
COM.ONIXSUITE.9782930344782
03
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Presses universitaires de Louvain
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SKU
70957
02
2930344784
03
9782930344782
15
9782930344782
10
BC
<TitleType>01</TitleType>
<TitleText>Thèses de l'Université catholique de Louvain (UCL)</TitleText>
Numéro 2
Thèses de la Faculté des sciences
2
<TitleType>01</TitleType>
<TitleText>Generalised algebraic models</TitleText>
01
GCOI
29303100658430
1
A01
Claudia Centazzo
Centazzo, Claudia
Claudia
Centazzo
1
01
eng
194
00
194
03
SCI000000
29
2012
3051
SCIENCES FONDAMENTALES
24
INTERNET
Algèbre, géometrie et logique
24
INTERNET
Mathématiques
93
P
01
06
01
<P><BR>Algebraic theories and algebraic categories offer an innovative and revelatory description of the syntax and the semantics. An <EM>algebraic theory</EM> 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 <EM>algebraic category</EM>. 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 l<EM>ocalizations </EM>(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 <EM>geometric morphisms</EM> (namely, functors together with a finite limit-preserving left adjoint), in both the presheaf and the algebraic context.</P>
03
<P><BR>Algebraic theories and algebraic categories offer an innovative and revelatory description of the syntax and the semantics. An <EM>algebraic theory</EM> 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 <EM>algebraic category</EM>. 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 l<EM>ocalizations </EM>(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 <EM>geometric morphisms</EM> (namely, functors together with a finite limit-preserving left adjoint), in both the presheaf and the algebraic context.</P>
02
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...
01
<P><BR>Algebraic theories and algebraic categories offer an innovative and revelatory description of the syntax and the semantics. An <EM>algebraic theory</EM> 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 <EM>algebraic category</EM>. 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 l<EM>ocalizations </EM>(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 <EM>geometric morphisms</EM> (namely, functors together with a finite limit-preserving left adjoint), in both the presheaf and the algebraic context.</P>
03
<P><BR>Algebraic theories and algebraic categories offer an innovative and revelatory description of the syntax and the semantics. An <EM>algebraic theory</EM> 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 <EM>algebraic category</EM>. 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 l<EM>ocalizations </EM>(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 <EM>geometric morphisms</EM> (namely, functors together with a finite limit-preserving left adjoint), in both the presheaf and the algebraic context.</P>
02
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...
04
<p>Introduction 1</P><p>1 The theory of locally D-presentable categories 9</P><p>1.1 D-filtered categories . . . . . . . . . . . . . . . . . . . . . 9</P><p>- D-flat functors and sound doctrines . . . . . . . . . . . . 12</P><p>1.2 D-accessible categories . . . . . . . . . . . . . . . . . . . . 15</P><p>- The D-ind completion . . . . . . . . . . . . . . . . . . . 15</P><p>- The full subcategory of D-presentable objects . . . . . . 18</P><p>1.3 Locally D-presentable categories . . . . . . . . . . . . . . 22</P><p>- Restricting Yoneda . . . . . . . . . . . . . . . . . . . . . 25</P><p>1.4 D-completeness and Cauchy completeness . . . . . . . . . 26</P><p>2 The duality theorem for a limit doctrine 29</P><p>2.1 The duality Theorem . . . . . . . . . . . . . . . . . . . . . 29</P><p>- The 2-functor " . . . . . . . . . . . . . . . . . . . . . . . 29</P><p>- The objects of D-cont[A, set] . . . . . . . . . . . . . . . 32</P><p>- The 2-functor " is a biequivalence . . . . . . . . . . . . . 35</P><p>2.2 Some applications . . . . . . . . . . . . . . . . . . . . . . 39</P><p>- Comparing sound doctrines . . . . . . . . . . . . . . . . 39</P><p>2.3 A characterisation in terms of strong generators . . . . . . 54</P><p>- The Dop-colimit completion . . . . . . . . . . . . . . . . 55</P><p>- Strong generators play an important role . . . . . . . . . 56</P><p>- Consequences . . . . . . . . . . . . . . . . . . . . . . . . 62</P><p>3 On geometric morphisms and localizations of presheaf categories 69</P><p>3.1 Rambling about the concept of localization . . . . . . . . 69</P><p>- Exact and/or extensive categories . . . . . . . . . . . . . 70</P><p>- The exact completion of Fam . . . . . . . . . . . . . . . 75</P><p>- Some conditions our functors might satisfy . . . . . . . . 83</P><p><p></P>3.2 The localizations of presheaf categories . . . . . . . . . . . 87</P><p>- Comparing conditions . . . . . . . . . . . . . . . . . . . 94</P><p>- A sheaf-theoretical proof . . . . . . . . . . . . . . . . . . 98</P><p>4 On geometric morphisms and localizations of algebraic categories 121</P><p>4.1 Algebraic theories and algebraic categories . . . . . . . . . 121</P><p>4.2 A miscellany of preliminary results . . . . . . . . . . . . . 125</P><p>4.3 Some more conditions our functors might satisfy . . . . . 127</P><p>4.4 The coproduct completion and the exact completion in the case of an algebraic theory . .. 131</P><p>4.5 The localizations of (one sorted) algebraic categories . . . 152</P><p>4.6 The link with other characterisation theorems for localizations of algebraic categories . . 166</P><p>Acknowledgements 177</P><p>Bibliography 181</P>
23
01
https://pul.uclouvain.be/resources/titles/29303100658430/extras/70957_sciences_centazzov3_1002422.pdf
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BE
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https://pul.uclouvain.be/book/?GCOI=29303100658430
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3052405007518
Presses universitaires de Louvain
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Presses universitaires de Louvain
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