An introduction to fibrations, topos theory, the effective topos and modest sets:
Abstract: "A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical 'universe' of recursive mathematics. Among its objects are the modest sets, which form a set-theoretic model for polymorphism. More precisely, there is a fibrati...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Edinburgh
1992
|
| Schriftenreihe: | Laboratory for Foundations of Computer Science <Edinburgh>: LFCS report series
208 |
| Schlagworte: | |
| Zusammenfassung: | Abstract: "A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical 'universe' of recursive mathematics. Among its objects are the modest sets, which form a set-theoretic model for polymorphism. More precisely, there is a fibration of modest sets which satisfies suitable categorical completeness properties, that make it a model for various polymorphic type theories. These lecture notes provide a reasonably thorough introduction to this body of material, aimed at theoretical computer scientists rather than topos theorists. Chapter 2 is an outline of the theory of fibrations, and sketches how they can be used to model various typed [lambda]-calculi Chapter 3 is an exposition of some basic topos theory, and explains why a topos can be regarded as a model of set theory. Chapter 4 discusses the classical PER model for polymorphism, and shows how it 'lives inside' a particular topos -- the effective topos -- as the category of modest sets. An appendix contains a full presentation of the internal language of a topos, and a map of the effective topos. Chapters 2 and 3 provide a sampler of categorical type theory and categorical logic, and should be of more general interest than Chapter 4. They can be read more or less independently of each other; a connection is made at the end of Chapter 3 The main prerequisite for reading these notes is some basic category theory: limits and colimits, functors and natural transformations, adjoints, cartesian closed categories. No knowledge of indexed categories or categorical logic is needed. Some familiarity with 'ordinary' logic and typed [lambda]-calculus is assumed. |
| Beschreibung: | 150 S: |
Internformat
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| 490 | 1 | |a Laboratory for Foundations of Computer Science <Edinburgh>: LFCS report series |v 208 | |
| 520 | 3 | |a Abstract: "A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical 'universe' of recursive mathematics. Among its objects are the modest sets, which form a set-theoretic model for polymorphism. More precisely, there is a fibration of modest sets which satisfies suitable categorical completeness properties, that make it a model for various polymorphic type theories. These lecture notes provide a reasonably thorough introduction to this body of material, aimed at theoretical computer scientists rather than topos theorists. Chapter 2 is an outline of the theory of fibrations, and sketches how they can be used to model various typed [lambda]-calculi | |
| 520 | 3 | |a Chapter 3 is an exposition of some basic topos theory, and explains why a topos can be regarded as a model of set theory. Chapter 4 discusses the classical PER model for polymorphism, and shows how it 'lives inside' a particular topos -- the effective topos -- as the category of modest sets. An appendix contains a full presentation of the internal language of a topos, and a map of the effective topos. Chapters 2 and 3 provide a sampler of categorical type theory and categorical logic, and should be of more general interest than Chapter 4. They can be read more or less independently of each other; a connection is made at the end of Chapter 3 | |
| 520 | 3 | |a The main prerequisite for reading these notes is some basic category theory: limits and colimits, functors and natural transformations, adjoints, cartesian closed categories. No knowledge of indexed categories or categorical logic is needed. Some familiarity with 'ordinary' logic and typed [lambda]-calculus is assumed. | |
| 650 | 7 | |a Computer software |2 sigle | |
| 650 | 7 | |a Mathematics |2 sigle | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Lambda calculus | |
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Datensatz im Suchindex
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| any_adam_object | |
| author | Phoa, Wesley |
| author_facet | Phoa, Wesley |
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| author_sort | Phoa, Wesley |
| author_variant | w p wp |
| building | Verbundindex |
| bvnumber | BV009692778 |
| classification_tum | DAT 510f DAT 189f |
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| discipline | Informatik |
| format | Book |
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| id | DE-604.BV009692778 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:39:18Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006410362 |
| oclc_num | 28056605 |
| open_access_boolean | |
| physical | 150 S: |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| record_format | marc |
| series | Laboratory for Foundations of Computer Science <Edinburgh>: LFCS report series |
| series2 | Laboratory for Foundations of Computer Science <Edinburgh>: LFCS report series |
| spelling | Phoa, Wesley Verfasser aut An introduction to fibrations, topos theory, the effective topos and modest sets by Wesley Phoa Edinburgh 1992 150 S: txt rdacontent n rdamedia nc rdacarrier Laboratory for Foundations of Computer Science <Edinburgh>: LFCS report series 208 Abstract: "A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical 'universe' of recursive mathematics. Among its objects are the modest sets, which form a set-theoretic model for polymorphism. More precisely, there is a fibration of modest sets which satisfies suitable categorical completeness properties, that make it a model for various polymorphic type theories. These lecture notes provide a reasonably thorough introduction to this body of material, aimed at theoretical computer scientists rather than topos theorists. Chapter 2 is an outline of the theory of fibrations, and sketches how they can be used to model various typed [lambda]-calculi Chapter 3 is an exposition of some basic topos theory, and explains why a topos can be regarded as a model of set theory. Chapter 4 discusses the classical PER model for polymorphism, and shows how it 'lives inside' a particular topos -- the effective topos -- as the category of modest sets. An appendix contains a full presentation of the internal language of a topos, and a map of the effective topos. Chapters 2 and 3 provide a sampler of categorical type theory and categorical logic, and should be of more general interest than Chapter 4. They can be read more or less independently of each other; a connection is made at the end of Chapter 3 The main prerequisite for reading these notes is some basic category theory: limits and colimits, functors and natural transformations, adjoints, cartesian closed categories. No knowledge of indexed categories or categorical logic is needed. Some familiarity with 'ordinary' logic and typed [lambda]-calculus is assumed. Computer software sigle Mathematics sigle Mathematik Lambda calculus Set theory Laboratory for Foundations of Computer Science <Edinburgh>: LFCS report series 208 (DE-604)BV008930032 208 |
| spellingShingle | Phoa, Wesley An introduction to fibrations, topos theory, the effective topos and modest sets Laboratory for Foundations of Computer Science <Edinburgh>: LFCS report series Computer software sigle Mathematics sigle Mathematik Lambda calculus Set theory |
| title | An introduction to fibrations, topos theory, the effective topos and modest sets |
| title_auth | An introduction to fibrations, topos theory, the effective topos and modest sets |
| title_exact_search | An introduction to fibrations, topos theory, the effective topos and modest sets |
| title_full | An introduction to fibrations, topos theory, the effective topos and modest sets by Wesley Phoa |
| title_fullStr | An introduction to fibrations, topos theory, the effective topos and modest sets by Wesley Phoa |
| title_full_unstemmed | An introduction to fibrations, topos theory, the effective topos and modest sets by Wesley Phoa |
| title_short | An introduction to fibrations, topos theory, the effective topos and modest sets |
| title_sort | an introduction to fibrations topos theory the effective topos and modest sets |
| topic | Computer software sigle Mathematics sigle Mathematik Lambda calculus Set theory |
| topic_facet | Computer software Mathematics Mathematik Lambda calculus Set theory |
| volume_link | (DE-604)BV008930032 |
| work_keys_str_mv | AT phoawesley anintroductiontofibrationstopostheorytheeffectivetoposandmodestsets |