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Category Theory

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 ... More


Triangulated Categories

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199296866
eISBN:
9780191711329
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296866.003.0001
Subject:
Mathematics, Geometry / Topology

Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria ... More


Fibre Functors

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0004
Subject:
Mathematics, Number Theory

This chapter introduces the fiber functor suggested by Pierre Deligne. The proof that it is in fact a fiber functor is given in the Appendix.


Appendix: Deligne’s Fibre Functor

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0031
Subject:
Mathematics, Number Theory

This chapter takes up the proof of Theorem 3.1. It shows that N ↦ ω‎(N) := H⁰(𝔸¹/k¯, j0!N) is a fiber functor on the Tannakian category Ƿsubscript geom of those perverse sheaves on 𝔾ₘ/k¯ satisfying ... More


FUNCTORS AND NATURALITY

Steve Awodey

in Category Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198568612
eISBN:
9780191717567
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568612.003.0007
Subject:
Mathematics, Algebra

This chapter develops a general theory for functors. Topics discussed include category of categories, representable structure, stone duality, naturality, examples of natural transformations, ... More


CATEGORIES OF DIAGRAMS

Steve Awodey

in Category Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198568612
eISBN:
9780191717567
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568612.003.0008
Subject:
Mathematics, Algebra

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it ... More


ADJOINTS

Steve Awodey

in Category Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198568612
eISBN:
9780191717567
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568612.003.0009
Subject:
Mathematics, Algebra

This chapter focuses on notion of adjoint functor, which applies everything that has been learned so far to unify and subsume all the different universal mapping properties encountered, from free ... More


INTERACTIVE PROGRAMS AND WEAKLY FINAL COALGEBRAS IN DEPENDENT TYPE THEORY

Peter Hancock and Anton Setzer

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0007
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter considers the representation of interactive programs in dependent type theory and bases it on the monadic I/O used in Haskell. Two versions are described: in the first the interface with ... More


‘There is no Ontology Here’: Visual and Structural Geometry in Arithmetic

Colin McLarty

in The Philosophy of Mathematical Practice

Published in print:
2008
Published Online:
February 2010
ISBN:
9780199296453
eISBN:
9780191711961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296453.003.0015
Subject:
Philosophy, Logic/Philosophy of Mathematics

Today's number theory solves classical problems by structural tools that violate standard philosophical expectations in ontology. The far-reaching practical demands of this mathematics require on one ... More


Miscellaneous Strategies

John P. Burgess and Gideon Rosen

in A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250128
eISBN:
9780191597138
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250126.003.0006
Subject:
Philosophy, Logic/Philosophy of Mathematics

Indicates very briefly a variety of further strategies, beginning with modifications and combinations of the three that were discussed in the preceding three chapters. The more exotic strategies are ... More


An equivalence of categories

Ehud Hrushovski and François Loeser

in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.003.0013
Subject:
Mathematics, Geometry / Topology

This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable ... More


2-Dimensional Categories

Niles Johnson and Donald Yau

Published in print:
2021
Published Online:
February 2021
ISBN:
9780198871378
eISBN:
9780191914850
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198871378.001.0001
Subject:
Mathematics, Geometry / Topology

2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a ... More


Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'

Olivia Caramello

Published in print:
2017
Published Online:
March 2018
ISBN:
9780198758914
eISBN:
9780191818752
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198758914.001.0001
Subject:
Mathematics, Geometry / Topology, Pure Mathematics

This book is devoted to a general study of geometric theories from a topos-theoretic perspective. After recalling the necessary topos-theoretic preliminaries, it presents the main methodology it uses ... More


Ontological Nihilism

Jason Turner

in Oxford Studies in Metaphysics: volume 6

Published in print:
2011
Published Online:
May 2011
ISBN:
9780199603039
eISBN:
9780191725418
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199603039.003.0001
Subject:
Philosophy, Metaphysics/Epistemology, General

Ontological nihilism is the radical-sounding thesis that there is nothing at all. This chapter first discusses how the most plausible forms of this thesis aim to be slightly less radical than they ... More


Monad combinatorics

Ash Asudeh and Gianluca Giorgolo

in Enriched Meanings: Natural Language Semantics with Category Theory

Published in print:
2020
Published Online:
November 2020
ISBN:
9780198847854
eISBN:
9780191882470
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198847854.003.0007
Subject:
Linguistics, Semantics and Pragmatics

This chapter considers how the monadic formalization deals with interactions between the phenomena from the second part of the book by examining the pairwise interaction of all three phenomena. ... More


Categories

Niles Johnson and Donald Yau

in 2-Dimensional Categories

Published in print:
2021
Published Online:
February 2021
ISBN:
9780198871378
eISBN:
9780191914850
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198871378.003.0001
Subject:
Mathematics, Geometry / Topology

In this chapter, categories are defined, and basic concepts are reviewed. Starting from the definitions of a category, a functor, and a natural transformation, the chapter reviews limits, ... More


Functors, Transformations, and Modifications

Niles Johnson and Donald Yau

in 2-Dimensional Categories

Published in print:
2021
Published Online:
February 2021
ISBN:
9780198871378
eISBN:
9780191914850
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198871378.003.0004
Subject:
Mathematics, Geometry / Topology

This chapter discusses functors, transformations, and modifications that are bicategorical analogs of functors and natural transformations. The main concepts covered are lax functors, lax ... More


Diagrams of Variation: Functor Categories and Presheaves

Rocco Gangle

in Diagrammatic Immanence: Category Theory and Philosophy

Published in print:
2015
Published Online:
September 2016
ISBN:
9781474404174
eISBN:
9781474418645
Item type:
chapter
Publisher:
Edinburgh University Press
DOI:
10.3366/edinburgh/9781474404174.003.0005
Subject:
Philosophy, Metaphysics/Epistemology

This chapter begins by posing the question of linguistic meaning within the context of the mathematics of category theory. It then explains the notions of functor categories, natural transformations ... More


A Category Theory Explanation for Systematicity: Universal Constructions

Steven Phillips and William H. Wilson

in The Architecture of Cognition: Rethinking Fodor and Pylyshyn's Systematicity Challenge

Published in print:
2014
Published Online:
September 2014
ISBN:
9780262027236
eISBN:
9780262322461
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262027236.003.0009
Subject:
Philosophy, Philosophy of Mind

A theory of cognitive architecture should explain why systematicity is a necessary, not just possible consequence of the theory's core principles and base assumptions, without relying on arbitrary ... More


Introduction

Brian Conrad and Gopal Prasad

in Classification of Pseudo-reductive Groups (AM-191)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167923
eISBN:
9781400874026
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167923.003.0001
Subject:
Mathematics, Numerical Analysis

This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the ... More


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