*Steve Awodey*

- 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 ...
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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 is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.Less

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 is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.

*Niles Johnson and Donald Yau*

- Published in print:
- 2021
- Published Online:
- February 2021
- ISBN:
- 9780198871378
- eISBN:
- 9780191914850
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198871378.003.0008
- Subject:
- Mathematics, Geometry / Topology

In this chapter, the Yoneda Lemma and the Coherence Theorem for bicategories are stated and proved. The chapter discusses the bicategorical Yoneda pseudofunctor, a bicategorical version of the Yoneda ...
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In this chapter, the Yoneda Lemma and the Coherence Theorem for bicategories are stated and proved. The chapter discusses the bicategorical Yoneda pseudofunctor, a bicategorical version of the Yoneda embedding for a bicategory, which is a local equivalence, and the Bicategorical Yoneda Lemma. A consequence of the Bicategorical Whitehead Theorem and the Bicategorical Yoneda Embedding is the Bicategorical Coherence Theorem, which states that every bicategory is biequivalent to a 2-category.Less

In this chapter, the Yoneda Lemma and the Coherence Theorem for bicategories are stated and proved. The chapter discusses the bicategorical Yoneda pseudofunctor, a bicategorical version of the Yoneda embedding for a bicategory, which is a local equivalence, and the Bicategorical Yoneda Lemma. A consequence of the Bicategorical Whitehead Theorem and the Bicategorical Yoneda Embedding is the Bicategorical Coherence Theorem, which states that every bicategory is biequivalent to a 2-category.

*Niles Johnson and Donald Yau*

- 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, ...
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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, adjunctions, equivalences, the Yoneda Lemma, monads, monoidal categories, and Mac Lane's Coherence Theorem. Enriched categories, which provide one characterization of 2-categories, are also discussed. This chapter makes this book self-contained and accessible to beginners.Less

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, adjunctions, equivalences, the Yoneda Lemma, monads, monoidal categories, and Mac Lane's Coherence Theorem. Enriched categories, which provide one characterization of 2-categories, are also discussed. This chapter makes this book self-contained and accessible to beginners.

*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 ...
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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 systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.Less

*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 systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.