🔢Category Theory Unit 13 – The Yoneda Lemma and Presheaves

Key Concepts and Definitions

• Category theory studies objects and morphisms between them in an abstract setting
• Functors map between categories preserving composition and identity morphisms
• Natural transformations provide a way to compare functors
• Yoneda Lemma establishes a bijection between the set of natural transformations from a representable functor to any other functor and the set of elements of that functor applied to the representing object
  • Representable functors are functors naturally isomorphic to a Hom-functor for some object in the category
• Presheaves are contravariant functors from a category to the category of sets
  • They can be thought of as a generalization of the concept of a sheaf
• The category of presheaves on a category $\mathcal{C}$, denoted $\mathbf{Set}^{\mathcal{C}^{op}}$, has presheaves as objects and natural transformations between them as morphisms

Historical Context and Development

• Category theory emerged in the 1940s through the work of Samuel Eilenberg and Saunders Mac Lane
• Initially developed as a tool for algebraic topology, it has since found applications in various branches of mathematics
• The Yoneda Lemma, named after Nobuo Yoneda, was introduced in the 1950s
  • It is considered one of the most fundamental results in category theory
• Presheaves were introduced as a generalization of sheaves, which were studied in algebraic geometry
• The concept of presheaves has been applied to various areas, including algebraic topology, algebraic geometry, and logic
  • Presheaves play a crucial role in the development of topos theory

The Yoneda Lemma: Statement and Intuition

• The Yoneda Lemma states that for any locally small category $\mathcal{C}$ and any object $A$ in $\mathcal{C}$, there is a bijection between the set of natural transformations from the Hom-functor $\text{Hom}(A, -)$ to any other functor $F: \mathcal{C} \to \mathbf{Set}$ and the set $F(A)$
  • In other words, $\text{Nat}(\text{Hom}(A, -), F) \cong F(A)$
• The bijection is given by evaluating a natural transformation at the identity morphism of $A$
• Intuitively, the Yoneda Lemma tells us that an object $A$ in a category $\mathcal{C}$ can be completely determined by the morphisms into it from other objects in the category
  • This provides a way to embed a category into the category of presheaves on it
• The Yoneda embedding, which sends an object $A$ to the Hom-functor $\text{Hom}(-, A)$, is a full and faithful functor

Understanding Presheaves

• A presheaf on a category $\mathcal{C}$ is a contravariant functor $F: \mathcal{C}^{op} \to \mathbf{Set}$
  • For each object $A$ in $\mathcal{C}$, it assigns a set $F(A)$
  • For each morphism $f: A \to B$ in $\mathcal{C}$, it assigns a function $F(f): F(B) \to F(A)$ in the opposite direction
• Presheaves can be thought of as a way to assign "local data" to each object in a category
  • The morphisms in the category allow us to relate the local data between different objects
• The category of presheaves on $\mathcal{C}$, denoted $\mathbf{Set}^{\mathcal{C}^{op}}$, has presheaves as objects and natural transformations between them as morphisms
  • This category has nice properties, such as being complete, cocomplete, and cartesian closed
• Presheaves are used in various contexts, such as defining sheaves in algebraic geometry and constructing topoi in topos theory

Connections Between Yoneda Lemma and Presheaves

• The Yoneda Lemma provides a way to embed a category $\mathcal{C}$ into the category of presheaves on it, $\mathbf{Set}^{\mathcal{C}^{op}}$, via the Yoneda embedding
  • The Yoneda embedding sends an object $A$ to the representable presheaf $\text{Hom}(-, A)$
• The Yoneda Lemma states that the Yoneda embedding is fully faithful, meaning it preserves and reflects morphisms
  • This implies that a category can be studied by looking at its presheaf category
• The Yoneda Lemma can be used to prove various properties of presheaf categories, such as the existence of limits and colimits
• Presheaves provide a way to generalize the Yoneda Lemma to enriched categories
  • In this context, the Yoneda embedding takes values in the category of enriched presheaves

Applications and Examples

• The Yoneda Lemma is used to prove the representability of various functors in algebraic geometry
  • For example, the functor of points of a scheme is representable by the scheme itself
• In algebraic topology, the singular homology and cohomology functors can be defined using presheaves
  • The Yoneda Lemma is used to establish the representability of these functors
• Presheaves are used to define sheaves in algebraic geometry
  • A sheaf is a presheaf satisfying certain gluing conditions
  • The category of sheaves on a topological space forms a topos
• The Yoneda Lemma is used to establish the equivalence between the category of simplicial sets and the category of presheaves on the simplex category
  • This equivalence is fundamental in the study of homotopy theory
• In logic and theoretical computer science, presheaf categories are used to model dependent type theories and higher-order logic

Common Misconceptions and Pitfalls

• The Yoneda Lemma is often confused with the Yoneda embedding
  • The Yoneda Lemma is a statement about natural transformations, while the Yoneda embedding is a functor
• The contravariance of presheaves can be a source of confusion
  • It is important to keep track of the direction of morphisms when working with presheaves
• Presheaves are sometimes mistaken for sheaves
  • While every sheaf is a presheaf, not every presheaf is a sheaf
  • Sheaves satisfy additional gluing conditions
• The Yoneda Lemma is not limited to locally small categories
  • It can be generalized to enriched categories and even higher categories
• The Yoneda Lemma does not require the Axiom of Choice
  • However, some of its consequences, such as the existence of injective resolutions, may require the Axiom of Choice

Further Reading and Resources

• "Categories for the Working Mathematician" by Saunders Mac Lane is a classic textbook on category theory that covers the Yoneda Lemma and presheaves in detail
• "Sheaves in Geometry and Logic: A First Introduction to Topos Theory" by Saunders Mac Lane and Ieke Moerdijk provides a gentle introduction to presheaves and sheaves from a logical perspective
• "Sketches of an Elephant: A Topos Theory Compendium" by Peter Johnstone is a comprehensive treatise on topos theory, which heavily relies on presheaves and the Yoneda Lemma
• The nLab (https://ncatlab.org/) is an online wiki that contains a wealth of information on category theory, including articles on the Yoneda Lemma and presheaves
• The Stacks Project (https://stacks.math.columbia.edu/) is an online reference for algebraic geometry that includes a detailed treatment of presheaves and sheaves