1.2 Morphisms, isomorphisms, and functors

Category Theory 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

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

Isomorphisms and two-sided inverses

• Proof outline
  1. Assume $f$ is isomorphism
     • Definition provides $g$ with $g \circ f = 1_A$ and $f \circ g = 1_B$
     • $g$ serves as two-sided inverse of $f$
  2. Assume $f$ has two-sided inverse $g$
     • Given $g \circ f = 1_A$ and $f \circ g = 1_B$
     • 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

• Functor definition
  • Map $F: C \to D$ between categories $C$ and $D$ preserves categorical structure
  • Assigns objects to objects and morphisms to morphisms maintaining relationships
• Components of functor
  • Object assignment maps each object $A$ in $C$ to $F(A)$ in $D$
  • Morphism assignment takes each morphism $f$ in $C$ to $F(f)$ in $D$
• 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

• Composition preservation
  • For composable morphisms $f: A \to B$ and $g: B \to C$ in $C$
  • Ensures $F(g \circ f) = F(g) \circ F(f)$ maintaining operation structure
• Identity morphism preservation
  • For object $A$ in $C$, functor maps $F(1_A) = 1_{F(A)}$
  • 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