Steve Awodey
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.001.0001
- Subject:
- Mathematics, Algebra
This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible ...
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This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.Less
This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.
D. Huybrechts
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, ...
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This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.Less
This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.
Hagit Borer
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199263936
- eISBN:
- 9780191759017
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199263936.003.0006
- Subject:
- Linguistics, Theoretical Linguistics, Syntax and Morphology
This chapter returns to the general model, and offers a perspective on phrase structure, constituent structure, linearization and functors. While Chomsky's Bare Phrase Structure is adopted, the ...
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This chapter returns to the general model, and offers a perspective on phrase structure, constituent structure, linearization and functors. While Chomsky's Bare Phrase Structure is adopted, the assumption that external merge involves lexically listed items (complete with category) is rejected, and is replaced with the claim that external merge involves roots, otherwise unspecified for category, and functors. The nature of C-functors is discussed in detail, and they are argued to be defined them as fundamentally syntactic functions which divide the categorial space, defining a categorial space as their domain, and, obligatorily transitive, defining their complement set as a distinct categorial space (their Categorial Complement Set). The notation CX[Y] is thus proposed, for a C-functor which projects, defines, domain X and which defines its complement domain as Y. The role of S-functors, fundamentally semantic functions, in turn is defined relative to the semantic value which they assign to otherwise empty sets, which are subsequently projected as segments of Extended Projections (ExP-segments). Extended Projections, in turn, are defined relative to their shared Categorial Complement Set. A somewhat speculative discussion concerning adjunction and linearization ends this chapter.Less
This chapter returns to the general model, and offers a perspective on phrase structure, constituent structure, linearization and functors. While Chomsky's Bare Phrase Structure is adopted, the assumption that external merge involves lexically listed items (complete with category) is rejected, and is replaced with the claim that external merge involves roots, otherwise unspecified for category, and functors. The nature of C-functors is discussed in detail, and they are argued to be defined them as fundamentally syntactic functions which divide the categorial space, defining a categorial space as their domain, and, obligatorily transitive, defining their complement set as a distinct categorial space (their Categorial Complement Set). The notation CX[Y] is thus proposed, for a C-functor which projects, defines, domain X and which defines its complement domain as Y. The role of S-functors, fundamentally semantic functions, in turn is defined relative to the semantic value which they assign to otherwise empty sets, which are subsequently projected as segments of Extended Projections (ExP-segments). Extended Projections, in turn, are defined relative to their shared Categorial Complement Set. A somewhat speculative discussion concerning adjunction and linearization ends this chapter.
Hagit Borer
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199263936
- eISBN:
- 9780191759017
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199263936.003.0013
- Subject:
- Linguistics, Theoretical Linguistics, Syntax and Morphology
This chapter provides a summary of the major results presented in the book, as well as an overview of their significance, particularly in light of past and future research agendas. A summary is ...
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This chapter provides a summary of the major results presented in the book, as well as an overview of their significance, particularly in light of past and future research agendas. A summary is presented of the syntactic, semantic, and phonological properties of the main building blocks proposed in this book: roots, C-functors, S-functors, and Extended Projections. In view of the conclusions reached regarding the role of phonological indices and phonological realization in the syntactic derivation, this chapter urges a re-evaluation of the role of some aspects of phonology in the syntactic derivation. It specifically questions the claim that is sometimes put forward that phonology as a whole cannot be part of narrow syntax.Less
This chapter provides a summary of the major results presented in the book, as well as an overview of their significance, particularly in light of past and future research agendas. A summary is presented of the syntactic, semantic, and phonological properties of the main building blocks proposed in this book: roots, C-functors, S-functors, and Extended Projections. In view of the conclusions reached regarding the role of phonological indices and phonological realization in the syntactic derivation, this chapter urges a re-evaluation of the role of some aspects of phonology in the syntactic derivation. It specifically questions the claim that is sometimes put forward that phonology as a whole cannot be part of narrow syntax.
Rocco Gangle
- Published in print:
- 2015
- Published Online:
- September 2016
- ISBN:
- 9781474404174
- eISBN:
- 9781474418645
- Item type:
- chapter
- Publisher:
- Edinburgh University Press
- DOI:
- 10.3366/edinburgh/9781474404174.003.0003
- Subject:
- Philosophy, Metaphysics/Epistemology
This chapter introduces the elementary concepts and operations of category theory. It introduces the notion of category by generalizing from partial orders and systems of functions and then ...
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This chapter introduces the elementary concepts and operations of category theory. It introduces the notion of category by generalizing from partial orders and systems of functions and then introduces the constructions of universal mapping properties and functors. It concludes by surveying a number of common examples.Less
This chapter introduces the elementary concepts and operations of category theory. It introduces the notion of category by generalizing from partial orders and systems of functions and then introduces the constructions of universal mapping properties and functors. It concludes by surveying a number of common examples.
Rocco Gangle
- Published in print:
- 2015
- Published Online:
- September 2016
- ISBN:
- 9781474404174
- eISBN:
- 9781474418645
- Item type:
- chapter
- Publisher:
- Edinburgh University Press
- DOI:
- 10.3366/edinburgh/9781474404174.003.0007
- Subject:
- Philosophy, Metaphysics/Epistemology
This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several ...
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This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several concrete examples. Topoi are introduced conceptually by focusing on the role of the subobject classifier within a topos, with examples from set theory and graph theory. The difference between the classical logic of Boolean algebras and the non-classical logic of Heyting algebras is explained in terms of their natural mathematical environments of set theory and topos theory respectively.Less
This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several concrete examples. Topoi are introduced conceptually by focusing on the role of the subobject classifier within a topos, with examples from set theory and graph theory. The difference between the classical logic of Boolean algebras and the non-classical logic of Heyting algebras is explained in terms of their natural mathematical environments of set theory and topos theory respectively.
Kai-Wen Lan
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691156545
- eISBN:
- 9781400846016
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156545.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter elaborates on the representability of the moduli problems defined in the previous chapter. The treatment here is biased towards the prorepresentability of local moduli and Artin's ...
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This chapter elaborates on the representability of the moduli problems defined in the previous chapter. The treatment here is biased towards the prorepresentability of local moduli and Artin's criterion of algebraic stacks. The geometric invariant theory or the theory of Barsotti–Tate groups has been set aside: the argument is very elementary and might be considered outdated by the experts in this area. The chapter, however, discusses the Kodaira–Spencer morphisms of abelian schemes with PEL structures, which are best understood via the study of deformation theory. It also considers the proof of the formal smoothness of local moduli functors, illustrating how the linear algebraic assumptions are used.Less
This chapter elaborates on the representability of the moduli problems defined in the previous chapter. The treatment here is biased towards the prorepresentability of local moduli and Artin's criterion of algebraic stacks. The geometric invariant theory or the theory of Barsotti–Tate groups has been set aside: the argument is very elementary and might be considered outdated by the experts in this area. The chapter, however, discusses the Kodaira–Spencer morphisms of abelian schemes with PEL structures, which are best understood via the study of deformation theory. It also considers the proof of the formal smoothness of local moduli functors, illustrating how the linear algebraic assumptions are used.
Hagit Borer
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199263936
- eISBN:
- 9780191759017
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199263936.003.0001
- Subject:
- Linguistics, Theoretical Linguistics, Syntax and Morphology
This chapter provides an introduction to the book as a whole. It begins with a detailed critical review of historical approaches to Words, focusing, in particular, on the rationale for listedness and ...
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This chapter provides an introduction to the book as a whole. It begins with a detailed critical review of historical approaches to Words, focusing, in particular, on the rationale for listedness and lexical word formation, and attempting to show that from both a phonological and syntactic perspective, that move is conceptually and theoretically problematic. This is followed by a preliminary presentation of the specifics of the system to be used throughout the book, eventually to be justified in detail in Chapters 6–10. Specifically, roots are introduced as a phonological index, and two distinct functors are introduced—a categorial functor (C-functor), responsible for the division of the categorial space, and a semantic functor (S-functor) responsible for valuing otherwise null terminals to project — once valued and thus categorized — as segments of Extended Projections. Extended Projections are likewise defined as a universally fixed set of segments of Extended Projections dominating a categorial core (C-core). The chapter ends with an outline and discussion of the general organization of the book.Less
This chapter provides an introduction to the book as a whole. It begins with a detailed critical review of historical approaches to Words, focusing, in particular, on the rationale for listedness and lexical word formation, and attempting to show that from both a phonological and syntactic perspective, that move is conceptually and theoretically problematic. This is followed by a preliminary presentation of the specifics of the system to be used throughout the book, eventually to be justified in detail in Chapters 6–10. Specifically, roots are introduced as a phonological index, and two distinct functors are introduced—a categorial functor (C-functor), responsible for the division of the categorial space, and a semantic functor (S-functor) responsible for valuing otherwise null terminals to project — once valued and thus categorized — as segments of Extended Projections. Extended Projections are likewise defined as a universally fixed set of segments of Extended Projections dominating a categorial core (C-core). The chapter ends with an outline and discussion of the general organization of the book.
Hagit Borer
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199263936
- eISBN:
- 9780191759017
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199263936.003.0007
- Subject:
- Linguistics, Theoretical Linguistics, Syntax and Morphology
This chapter is devoted to articulating a model of categorial determination based fundamentally on the conceptualization of categorial selection as a partition of the categorial space, and as ...
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This chapter is devoted to articulating a model of categorial determination based fundamentally on the conceptualization of categorial selection as a partition of the categorial space, and as establishing equivalence classes. In essence, functors define not only the category which they, themselves project, but also define a complement categorial domain, which comes to be associated with their complements. If such complements are otherwise category-less (e.g. roots) they thus come to be equivalent to a category (e.g. V-equivalent, N-equivalent and so on). If the complement is already categorial (e.g. itself headed by a functor), the existence of a complement categorial domain amounts, effectively, to a checking or a selection mechanism ruling out, e.g. the merger of a V-selecting functor such as -ation with a derived adjective such as ‘available’. Crucially, the model of categorization put forth is committed to the categorization of form in ‘the form’ as N or of form within ‘formation’ as V without the presence of additional structure, i.e., in both these cases ‘form’ is crucially a terminal and mono-morphemic. As a consequence, the account is committed to the absence of zero-affixes marking ‘form’ as N or V respectively. Much of the chapter, consequently, is devoted to arguing against the existence of zero-instantiated C-functors in English. Final comments concern the status of multi-function functors such as -ing and the status of adjectives.Less
This chapter is devoted to articulating a model of categorial determination based fundamentally on the conceptualization of categorial selection as a partition of the categorial space, and as establishing equivalence classes. In essence, functors define not only the category which they, themselves project, but also define a complement categorial domain, which comes to be associated with their complements. If such complements are otherwise category-less (e.g. roots) they thus come to be equivalent to a category (e.g. V-equivalent, N-equivalent and so on). If the complement is already categorial (e.g. itself headed by a functor), the existence of a complement categorial domain amounts, effectively, to a checking or a selection mechanism ruling out, e.g. the merger of a V-selecting functor such as -ation with a derived adjective such as ‘available’. Crucially, the model of categorization put forth is committed to the categorization of form in ‘the form’ as N or of form within ‘formation’ as V without the presence of additional structure, i.e., in both these cases ‘form’ is crucially a terminal and mono-morphemic. As a consequence, the account is committed to the absence of zero-affixes marking ‘form’ as N or V respectively. Much of the chapter, consequently, is devoted to arguing against the existence of zero-instantiated C-functors in English. Final comments concern the status of multi-function functors such as -ing and the status of adjectives.
Hagit Borer
- Published in print:
- 2014
- Published Online:
- January 2015
- ISBN:
- 9780199665266
- eISBN:
- 9780191748554
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199665266.003.0006
- Subject:
- Linguistics, Syntax and Morphology, Psycholinguistics / Neurolinguistics / Cognitive Linguistics
This chapter focuses on the categorial properties of roots and proposes that, just like event structure, these emerge in the context of particular functional structure and as a consequence of it. For ...
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This chapter focuses on the categorial properties of roots and proposes that, just like event structure, these emerge in the context of particular functional structure and as a consequence of it. For Borer, functors, whether segments of extended projections or derivational categorizers, are viewed as elements that partition the categorial space. Borer explicitly argues against linking the emergence of a category to zero-realized n, v, and a.Less
This chapter focuses on the categorial properties of roots and proposes that, just like event structure, these emerge in the context of particular functional structure and as a consequence of it. For Borer, functors, whether segments of extended projections or derivational categorizers, are viewed as elements that partition the categorial space. Borer explicitly argues against linking the emergence of a category to zero-realized n, v, and a.
