The Yoneda Lemma and Coherence

2021 ◽  
pp. 305-330
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Coherence Theorem ◽  
Local Equivalence ◽  
Yoneda Lemma

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.

2-Dimensional Categories

2021 ◽  
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Tensor Product ◽  
Category Theory ◽  
Basic Category ◽  
Double Categories ◽  
Grothendieck Construction ◽  
Monoidal Bicategories ◽  
Coherence Theorem ◽  
First Time ◽  
Yoneda Lemma

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.

Categories

2021 ◽  
pp. 1-34
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Natural Transformation ◽  
Monoidal Categories ◽  
Enriched Categories ◽  
Basic Concepts ◽  
Coherence Theorem ◽  
Yoneda Lemma

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.

Flat functors and classifying toposes

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
General Theory ◽  
Geometric Theory ◽  
Small Category ◽  
Independent Interest ◽  
Yoneda Lemma ◽  
Geometric Morphisms ◽  
The Way

This chapter develops a general theory of extensions of flat functors along geometric morphisms of toposes; the attention is focused in particular on geometric morphisms between presheaf toposes induced by embeddings of categories and on geometric morphisms to the classifying topos of a geometric theory induced by a small category of set-based models of the latter. A number of general results of independent interest are established on the way, including developments on colimits of internal diagrams in toposes and a way of representing flat functors by using a suitable internalized version of the Yoneda lemma. These general results will be instrumental for establishing in Chapter 6 the main theorem characterizing the class of geometric theories classified by a presheaf topos and for applying it.

scholarly journals Proof of a Conjecture of S. Mac Lane

1996 ◽  
Vol 3 (61) ◽  
Author(s):  
Sergei Soloviev
Keyword(s):  
Linear Logic ◽  
Sufficient Conditions ◽  
Vector Spaces ◽  
Categorical Equivalence ◽  
Closed Category ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category ◽  
Coherence Theorem ◽  
Mac Lane

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description.

Functors, Transformations, and Modifications

2021 ◽  
pp. 147-202
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Natural Transformations ◽  
Yoneda Lemma

This chapter discusses functors, transformations, and modifications that are bicategorical analogs of functors and natural transformations. The main concepts covered are lax functors, lax transformations, modifications, and icons. A section is devoted to representable pseudofunctors, representable transformations, and representable modifications, which will be used in the Bicategorical Yoneda Lemma.

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