Glossary

1.2 Morphisms, isomorphisms, and functors

3 min read•july 25, 2024

fundamentals are crucial building blocks for understanding more complex concepts in Topos Theory. This section introduces key morphisms like monomorphisms, epimorphisms, and isomorphisms, which generalize familiar concepts from set theory to broader categorical settings.

Functors, the focus of the next part, are essential tools for connecting different categories. They allow us to map objects and morphisms between categories while preserving important structural relationships, forming the basis for more advanced categorical constructions.

Category Theory Fundamentals

Monomorphisms, epimorphisms, and isomorphisms

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    • f:A→Bf: A \to B in category CC acts like injective function generalizes injectivity from Set theory
    • Left-cancellative property ensures for morphisms g,h:X→Ag, h: X \to A, f∘g=f∘hf \circ g = f \circ h implies g=hg = h
    • Examples: inclusion maps (subsets), group homomorphisms with trivial kernel
    • Morphism f:A→Bf: A \to B in category CC behaves like surjective function extends surjectivity concept
    • Right-cancellative property guarantees for morphisms g,h:B→Yg, h: B \to Y, g∘f=h∘fg \circ f = h \circ f implies g=hg = h
    • Examples: quotient maps (groups), projection maps (product spaces)
    • Morphism f:A→Bf: A \to B in category CC has inverse g:B→Ag: B \to A satisfying g∘f=1Ag \circ f = 1_A and f∘g=1Bf \circ g = 1_B
    • Generalizes bijective functions preserves structure between objects
    • Examples: vector space isomorphisms, group isomorphisms, homeomorphisms (topology)

Isomorphisms and two-sided inverses

  • Proof outline
    1. Assume ff is isomorphism
      • Definition provides gg with g∘f=1Ag \circ f = 1_A and f∘g=1Bf \circ g = 1_B
      • gg serves as two-sided inverse of ff
    2. Assume ff has two-sided inverse gg
      • Given g∘f=1Ag \circ f = 1_A and f∘g=1Bf \circ g = 1_B
      • Conditions match isomorphism definition
  • Key points
    • Inverse uniqueness ensures only one two-sided inverse exists
    • Isomorphisms possess both monomorphism and epimorphism properties
    • Examples: matrix inverses, function inverses (bijective functions)

Functors and Their Properties

Functors between categories

  • definition
    • Map F:C→DF: C \to D between categories CC and DD preserves categorical structure
    • Assigns objects to objects and morphisms to morphisms maintaining relationships
  • Components of functor
    • Object assignment maps each object AA in CC to F(A)F(A) in DD
    • Morphism assignment takes each morphism ff in CC to F(f)F(f) in DD
  • Examples of functors
    • Forgetful functor from Group to Set strips group structure
    • Power set functor from Set to Set maps sets to their power sets
    • Fundamental group functor from Top to Group associates topological spaces with groups
    • Constant functor maps all objects to single object and all morphisms to identity
    • Identity functor maps category to itself preserving all structure

Properties of functors

  • preservation
    • For composable morphisms f:A→Bf: A \to B and g:B→Cg: B \to C in CC
    • Ensures F(g∘f)=F(g)∘F(f)F(g \circ f) = F(g) \circ F(f) maintaining operation structure
  • preservation
    • For object AA in CC, functor maps F(1A)=1F(A)F(1_A) = 1_{F(A)}
    • Preserves identity elements across categories
  • Consequences of properties
    • Commutative diagrams remain commutative when mapped by functors
    • Isomorphisms transform into isomorphisms under functor application
  • Types of functors
    • Covariant functors preserve morphism direction (standard functors)
    • Contravariant functors reverse morphism direction (dual category relationship)
  • Functors and category structure
    • Maintain source and target object relationships for morphisms
    • Map domains to domains and codomains to codomains preserving overall structure

Key Terms to Review (18)

Adjoint Functors Theorem: The Adjoint Functors Theorem states that under certain conditions, a functor has both a left adjoint and a right adjoint, which are crucial in establishing relationships between different categories. This theorem reveals how morphisms and isomorphisms behave between categories when functors are applied, demonstrating the powerful interplay between different mathematical structures. It highlights how adjunctions can simplify complex constructions and ensure the preservation of certain properties across categories.
Associativity: Associativity is a fundamental property in mathematics that refers to the way in which the grouping of elements affects the outcome of binary operations. In the context of morphisms, isomorphisms, and functors, associativity ensures that when combining multiple morphisms or operations, the result remains consistent regardless of how the elements are grouped. This property is essential for maintaining structure and coherence in mathematical systems, as it allows for flexibility in how operations are performed without altering the final result.
Category Theory: Category theory is a mathematical framework that deals with abstract structures and relationships between them, focusing on the concept of objects and morphisms. It provides a way to formalize mathematical concepts across various fields, emphasizing the connections and mappings between different structures rather than their individual components. This abstraction is crucial for understanding complex relationships in mathematics, including transformations through functors, the properties of isomorphisms, and connections to logic and foundational mathematics.
Charles Pontryagin: Charles Pontryagin was a prominent Russian mathematician known for his contributions to topology and functional analysis, particularly in the development of Pontryagin duality. His work has important implications for the study of morphisms, isomorphisms, and functors, as it provides deep insights into the structure and behavior of topological groups and their duals.
Commutative Diagram: A commutative diagram is a visual representation of mathematical relationships between objects and morphisms, where any two paths in the diagram that connect the same two objects yield the same result when composed. This concept highlights the compatibility of morphisms and their compositional relationships, making it essential for understanding structures in category theory. In particular, commutative diagrams facilitate the exploration of morphisms, isomorphisms, functors, and exponential objects, revealing how these elements interact and maintain structural integrity.
Composition: Composition refers to the process of combining two morphisms in a category to form a new morphism. This operation is essential as it allows for the chaining of relationships between objects, facilitating the exploration of how different structures interact within the framework of categories. Composition must satisfy specific properties, such as associativity and the existence of identity morphisms, which are crucial for the overall coherence of categorical structures.
Contravariant Functor: A contravariant functor is a type of mapping between categories that reverses the direction of morphisms, taking objects from one category to another while flipping the arrows. This means that if there is a morphism from object A to object B in the original category, a contravariant functor will map these objects to another morphism going from the image of B back to the image of A. Understanding contravariant functors is crucial for grasping how relationships between different mathematical structures can be modeled and transformed.
Covariant Functor: A covariant functor is a type of mapping between categories that preserves the direction of morphisms. In simpler terms, if you have a morphism (or arrow) from one object to another in the first category, a covariant functor will map that morphism to a morphism between the corresponding objects in the second category, keeping the same direction. This concept ties into how we understand morphisms and isomorphisms, as well as how different types of functors interact with natural transformations and help us explore functor categories and the Yoneda lemma.
Epimorphism: An epimorphism is a type of morphism in category theory that can be thought of as a generalization of the concept of surjectivity in set theory. It is defined as a morphism \( f: A \to B \) such that for any two morphisms \( g_1, g_2: B \to C \), if \( g_1 \circ f = g_2 \circ f \), then it must follow that \( g_1 = g_2 \). This means that an epimorphism is a morphism that, in a sense, covers all of its target object and ensures the uniqueness of how morphisms can factor through it, linking closely to isomorphisms and the nature of functors in category theory.
Functor: A functor is a mathematical mapping between categories that preserves the structure of those categories, meaning it maps objects to objects and morphisms to morphisms in a way that respects the composition and identity of the categories. Functors play a crucial role in connecting different mathematical structures and help in defining various concepts such as natural transformations and limits.
Functoriality: Functoriality refers to the property of a functor that maps morphisms in one category to morphisms in another category in a way that preserves the structure of the categories. This means that if there is a morphism between objects in the first category, the functor will produce a corresponding morphism between the images of those objects in the second category, maintaining composition and identity. Functoriality connects various mathematical concepts and structures, illustrating how they interact through mappings.
Identity morphism: An identity morphism is a special type of morphism in category theory that serves as the 'do-nothing' arrow for each object in a category, meaning it maps an object to itself. Every object in a category has an associated identity morphism, and it acts as a neutral element with respect to composition of morphisms, reinforcing the structure and coherence within the category.
Isomorphism: An isomorphism is a special type of morphism in category theory that indicates a structural similarity between two objects, meaning there exists a bijective correspondence between them that preserves the categorical structure. This concept allows us to understand when two mathematical structures can be considered 'the same' in a categorical sense, as it connects to important ideas like special objects, functors, and adjoint relationships.
Monomorphism: A monomorphism is a morphism that is left-cancellable, meaning if two morphisms composed with it yield the same result, then those two morphisms must be the same. This concept connects closely to notions of injectivity in set theory, highlighting how monomorphisms can represent the inclusion of one structure into another while preserving distinctiveness.
Morphism: A morphism is a structure-preserving map between two objects in a category, reflecting the relationships and transformations that can occur within that context. It plays a central role in connecting objects and understanding how they interact, serving as the foundation for defining concepts like isomorphisms and functors, which enrich the framework of category theory.
Naturality: Naturality is a property of certain mathematical constructions, particularly in category theory, where a transformation or a morphism can be shown to commute with other structures in a natural way. It emphasizes that such transformations do not depend on arbitrary choices and behave consistently across different contexts, making them more universally applicable. In the realm of functors, natural transformations highlight how functorial relationships are maintained, while adjunctions illustrate naturality in the context of units and counits, showcasing their integral role in the structure of categories.
Samuel Eilenberg: Samuel Eilenberg was a prominent mathematician known for his foundational work in category theory, topology, and algebra, particularly in the context of algebraic topology and topos theory. His contributions significantly advanced the understanding of categories, functors, and adjunctions, which are crucial concepts in modern mathematics.
Yoneda Lemma: The Yoneda Lemma is a foundational result in category theory that relates functors to natural transformations, stating that every functor from a category to the category of sets can be represented by a set of morphisms from an object in that category. This lemma highlights the importance of morphisms and allows for deep insights into the structure of categories and functors.
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