🔢Category Theory Unit 6 – Equivalence of Categories

Key Concepts and Definitions

• Equivalence of categories captures the notion of two categories being "essentially the same" while allowing for differences in their specific objects and morphisms
• An equivalence between categories $\mathcal{C}$ and $\mathcal{D}$ consists of a pair of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ satisfying certain properties
  • The composition $F \circ G$ is naturally isomorphic to the identity functor $1_{\mathcal{D}}$ on $\mathcal{D}$
  • The composition $G \circ F$ is naturally isomorphic to the identity functor $1_{\mathcal{C}}$ on $\mathcal{C}$
• The functors $F$ and $G$ in an equivalence are called "quasi-inverses" of each other, generalizing the notion of inverse functions
• Natural isomorphisms play a crucial role in the definition of equivalence, allowing for a "weak" form of equality between functors
• Equivalence of categories is a weaker notion than isomorphism of categories, which requires strict equality of the compositions $F \circ G$ and $G \circ F$ with the respective identity functors

Motivation and Historical Context

• The concept of equivalence of categories arose from the need to compare categories that are not strictly isomorphic but share the same essential structure
• Equivalence allows for a more flexible notion of sameness between categories, capturing the idea that two categories can be "essentially the same" even if their objects and morphisms differ
• The development of equivalence of categories was influenced by the work of mathematicians such as Saunders Mac Lane and Samuel Eilenberg in the mid-20th century
• Equivalence of categories has become a fundamental tool in various branches of mathematics, including algebraic topology, algebraic geometry, and representation theory
• The notion of equivalence is particularly useful when studying categories up to isomorphism, as it allows for a more robust comparison of categories

Types of Equivalence

• Several types of equivalence exist in category theory, each capturing a different level of sameness between categories
• Adjoint equivalence is a type of equivalence where the functors $F$ and $G$ form an adjoint pair, satisfying certain universal properties
  • In an adjoint equivalence, the unit and counit of the adjunction are natural isomorphisms
• Morita equivalence is a type of equivalence used in the study of rings and modules, capturing the idea of two rings having equivalent module categories
• Derived equivalence is a type of equivalence used in the study of triangulated categories and derived categories, capturing the idea of two categories having equivalent derived categories
• Homotopy equivalence is a type of equivalence used in the study of topological spaces and homotopy theory, capturing the idea of two spaces being homotopy equivalent
• Each type of equivalence has its own specific properties and applications, tailored to the particular context in which it is used

Constructing Equivalences

• Constructing equivalences between categories often involves finding suitable functors $F$ and $G$ and proving that they satisfy the required properties
• One common approach to constructing equivalences is to start with a functor $F: \mathcal{C} \to \mathcal{D}$ and try to find a "quasi-inverse" functor $G: \mathcal{D} \to \mathcal{C}$
  • This involves showing that the compositions $F \circ G$ and $G \circ F$ are naturally isomorphic to the respective identity functors
• Another approach is to use the concept of adjoint functors, as an adjoint equivalence automatically satisfies the conditions for an equivalence
• In some cases, equivalences can be constructed by "localizing" a category with respect to a class of morphisms, resulting in a new category that is equivalent to the original one
• The construction of equivalences often relies on the specific properties of the categories involved and may require creative use of category-theoretic tools and techniques

Properties and Consequences

• Equivalence of categories preserves many important categorical properties, such as limits, colimits, and adjunctions
  • If two categories are equivalent, then a limit (or colimit) in one category corresponds to a limit (or colimit) in the other category via the equivalence functors
• Equivalent categories have isomorphic hom-sets between corresponding objects, meaning that the structure of morphisms is preserved up to isomorphism
• Equivalence of categories is an equivalence relation on the class of all categories, satisfying reflexivity, symmetry, and transitivity
• If two categories are equivalent, then their skeleton categories (obtained by collapsing isomorphic objects) are isomorphic
• Equivalent categories share many algebraic and geometric properties, making equivalence a powerful tool for studying categories up to isomorphism
• The concept of equivalence allows for a more flexible comparison of categories, enabling the transfer of results and insights between seemingly different mathematical contexts

Examples and Applications

• The category of finite-dimensional vector spaces over a field $k$ is equivalent to the category of finite-dimensional matrices over $k$
  • This equivalence allows for the study of linear algebra using both the language of vector spaces and matrices
• The category of compact Hausdorff spaces is equivalent to the opposite category of commutative $C^*$-algebras
  • This equivalence, known as Gelfand duality, plays a fundamental role in the study of operator algebras and functional analysis
• The category of affine schemes is equivalent to the opposite category of commutative rings
  • This equivalence is a cornerstone of modern algebraic geometry, allowing for the study of geometric objects using algebraic tools
• In homotopy theory, the homotopy category of CW complexes is equivalent to the homotopy category of topological spaces
  • This equivalence allows for the study of homotopy theory using the more tractable class of CW complexes
• Morita equivalence provides a way to compare rings with equivalent module categories, which has applications in various areas of algebra and representation theory

Common Misconceptions

• It is important to note that equivalence of categories does not imply that the categories are identical or isomorphic
  • Equivalent categories can have different objects and morphisms, but they share the same essential structure
• Equivalence of categories is not the same as isomorphism of categories, which is a stricter notion requiring the compositions $F \circ G$ and $G \circ F$ to be exactly equal to the respective identity functors
• The existence of an equivalence between two categories does not necessarily imply that the categories are equivalent in all respects
  • Some properties, such as concreteness or well-pointedness, may not be preserved under equivalence
• It is not always easy to determine whether two given categories are equivalent, as finding suitable functors and proving the required properties can be challenging
• Equivalence of categories is not the only way to compare categories; other notions, such as adjunctions or Quillen equivalences, may be more appropriate in certain contexts

Advanced Topics and Extensions

• The theory of 2-categories and bicategories extends the notion of equivalence to higher categorical structures
  • In a 2-category, equivalences are replaced by "adjoint equivalences" or "biequivalences," which take into account the 2-morphisms between functors
• The concept of $\infty$-categories, also known as weak $\infty$-categories or quasicategories, provides a framework for studying higher-dimensional categorical structures
  • In the setting of $\infty$-categories, equivalences are generalized to "categorical equivalences" or "weak equivalences"
• Homotopy type theory is a foundational approach to mathematics that combines ideas from type theory, homotopy theory, and category theory
  • In homotopy type theory, the notion of equivalence plays a central role, as types are considered up to homotopy equivalence
• The study of derived categories and triangulated categories has led to the development of "derived equivalences" and "triangulated equivalences," which capture the notion of equivalence in these more structured settings
• Topos theory is a branch of mathematics that studies generalized spaces called topoi, which can be seen as categorical models of set theory
  • In topos theory, the notion of equivalence is generalized to "geometric morphisms," which provide a way to compare different topoi